Electrode configuration for extended plasma confinement

ABSTRACT

Methods and systems are provided for plasma confinement utilizing various electrode and valve configurations. In one example, a device includes a first electrode positioned to define an outer boundary of an acceleration volume, a second electrode arranged coaxially with respect to the first electrode and positioned to define an inner boundary of the acceleration volume, at least one power supply to drive an electric current along a Z-pinch plasma column between the first second electrodes, and a set of valves to provide gas to the acceleration volume to fuel the Z-pinch plasma column, wherein an electron flow of the electric current is in a first direction from the second electrode to the first electrode. In additional or alternative examples, a shaping part is conductively connected to the second electrode to, in a presence of the gas, cause a gas breakdown of the gas to generate a sheared flow velocity profile.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.17/827,396, entitled “ELECTRODE CONFIGURATION FOR EXTENDED PLASMACONFINEMENT,” and filed on May 27, 2022, which claims priority to eachof U.S. Provisional Application No. 63/194,866, entitled “APPARATUS ANDMETHOD FOR EXTENDED PLASMA CONFINEMENT,” and filed on May 28, 2021, andU.S. Provisional Application No. 63/194,877, entitled “ELECTRODECONFIGURATION FOR EXTENDED PLASMA CONFINEMENT,” and filed on May 28,2021. The entire contents of each of the above-identified applicationsare hereby incorporated by reference for all purposes.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made, at least in part, with government support underGrant Nos. DE-AR001010 and DE-AR001260 awarded by the United StatesDepartment of Energy. The government has certain rights in theinvention.

BACKGROUND

Unless otherwise indicated herein, the recitations disclosed in thissection are not considered to be prior art to the claims in thisapplication and are not admitted to be prior art by inclusion in thissection.

Nuclear fusion is the process of combining two nuclei. If the two nucleiof elements with atomic numbers less than 26 [that is, with a loweratomic number than iron (Fe)] are fused, energy is released. The releaseof energy is due to a slight difference in mass between reactants andproducts of the fusion reaction (e.g., in high-temperature fusion plasmareactors), as governed by the expression E=mc².

Nuclear fusion holds the promise of effectively limitless energy withmore manageable waste products than some existing energy sources.

Controlled nuclear fusion in fusion plasmas, with plasma reactionssustained over extensive time periods, may be stymied by fast growingplasma instabilities. A viable approach to such controlled nuclearfusion (also below noted as “controlled fusion” or just “fusion,” asnoun or adjective indicating nuclear fusion related features and/orproperties) continues to be pursued through the study of differentplasma confinement approaches. Such approaches confer distinctadvantages across varying levels of scientific maturity.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other embodiments, features, and aspects of the presentinvention are considered in more detail in relation to the followingdescription of embodiments shown in the accompanying drawings, in which:

FIG. 1 illustrates modeling results of a Z pinch with no shearedvelocity flow, in accordance with at least one embodiment;

FIG. 2 illustrates modeling results of a sheared velocity flowstabilized Z pinch, in accordance with at least one embodiment;

FIG. 3 illustrates modeling results of a sheared velocity flowstabilized Z pinch, in accordance with at least one embodiment;

FIG. 4 illustrates modeling results of a sheared velocity flowstabilized Z pinch, in accordance with at least one embodiment;

FIG. 5 illustrates results of integrated Z pinch masses, in accordancewith at least one embodiment;

FIG. 6A shows an isometric view of a device to generate and maintain anazimuthally symmetric sheared ion velocity flow, in accordance with atleast one embodiment;

FIG. 6B shows a cross-sectional view of the device shown in FIG. 6A;

FIG. 7 shows a shaping part, in accordance with at least one embodiment;

FIG. 8 shows schematically a process of initiating and driving anazimuthally symmetric sheared flow for stabilization of Z-pinchdischarges, in accordance with at least one embodiment;

FIG. 9 shows a schematic diagram illustrating a device to generate andmaintain an azimuthally symmetric sheared ion velocity flow, inaccordance with at least one embodiment;

FIGS. 10A-10F show schematically a process of initiating and driving anazimuthally symmetric sheared flow for stabilization of Z-pinchdischarges for different anode/cathode configurations, in accordancewith at least one embodiment;

FIG. 11 illustrates normalized radial profiles of magnetic field densityand temperature in a Bennett equilibrium, in accordance with at leastone embodiment;

FIG. 12A shows an example time trace, in accordance with at least oneembodiment;

FIG. 12B shows an example time trace of integrated radial ion kineticenergy, normalized by initial magnetic energy, in accordance with atleast one embodiment;

FIG. 13A shows ideal five-moment, two-fluid (5M2F) results, inaccordance with at least one embodiment;

FIG. 13B shows an example time trace of integrated radial ion kineticenergy, normalized by initial magnetic energy, in accordance with atleast one embodiment;

FIG. 14 shows ideal 5M2F results with several perturbed wavenumbers, inaccordance with at least one embodiment;

FIG. 15 shows ideal 5M2F mode structures, in accordance with at leastone embodiment;

FIG. 16A shows ideal 5M2F results with stabilization by linear flow, inaccordance with at least one embodiment;

FIG. 16B shows ideal 5M2F results with stabilization by parabolicsheared flow, in accordance with at least one embodiment;

FIG. 17 shows mode structures in ideal 5M2F modeling with sheared flow,in accordance with at least one embodiment;

FIGS. 18A-18D show mode growth behavior in the 5M2F model with andwithout an initial phase shift in the perturbation, in accordance withat least one embodiment;

FIG. 19 shows momentum diffusivities in the FuZE-like equilibrium, inaccordance with at least one embodiment;

FIG. 20 shows corrected and uncorrected ion thermal diffusivities, inaccordance with at least one embodiment;

FIG. 21 shows instability growth rates, in accordance with at least oneembodiment;

FIG. 22 shows instability growth rates, in accordance with at least oneembodiment;

FIG. 23 shows ion density profiles for nonlinear simulations startingfrom FuZE-like equilibria, in accordance with at least one embodiment;and

FIG. 24 shows normalized ion inventory and thermal energy, in accordancewith at least one embodiment.

DETAILED DESCRIPTION

Embodiments of the present disclosure may be better understood byreferring to the following descriptions, which should be read inconjunction with the accompanying drawings of particular exemplaryembodiments. This description of the illustrated embodiment, set outbelow to enable one to build and use an implementation of the invention,is not intended to limit the invention, but to serve as a particularexample thereof. Those skilled in the art should appreciate that theymay readily use the conception and specific embodiments disclosed as abasis for modifying or designing other methods and systems for carryingout the same purposes of the present disclosure. Those skilled in theart should also understand that such equivalent assemblies do not departfrom the spirit and scope of the disclosure in its broadest form.

In some examples, typical power supply devices and methods arranged toform and sustain an axial Z-pinch current may be ill-suited or whollyunable to generate and sustain sufficient sheared velocity axial flowusable for stabilization of Z-pinch plasmas. Accordingly, describedherein are embodiments of devices and processes for generation andsustainment of sufficient sheared velocity flow in the transitionalmedium associated with boundary regions of Z-pinch discharges.

A fusion device based on the Z-pinch [e.g., U. Shumlak, B. A. Nelson, E.L. Claveau, E. G. Forbes, R. P. Golingo, M C. Hughes, R. J. Oberto, M P.Ross, and T R. Weber, “Increasing plasma parameters using sheared flowstabilization of a Z-pinch,” Phys. Plasmas 24, 055702 (2017); “Shumlak'17”; incorporated herein by reference] may be attractive since it wouldbe geometrically simple, inherently compact, and relatively low-cost.Some more recent publications [e.g., U. Shumlak, “Z-pinch fusion,” J.Appl. Phys. 127, 200901 (2020); published online: 27 May 2020; “Shumlak'20”; also incorporated by reference] further elaborate on sheared-flowstabilization (SFS) to produce an equilibrium Z pinch, which may sustaina compressed plasma state for durations appreciably longer than otherplasma timescales.

One area of sheared flow optimization pertains to augmented control ofboth neutral and ionized gas feeding into an evacuated volume of anacceleration volume of a Z-pinch apparatus. An example of a preexistingdevice may be found in FIG. 3 of Shumlak '20, illustrating a schematiccross sectional-view of a vacuum vessel of the Fusion Z-pinch Experiment(FuZE) SFS Z-pinch experimental device. In FIG. 3, one internal gas-puffvalve is arranged substantially at a middle axial position of theacceleration volume to provide a portion of chosen neutral filling gasthrough the “Inner Electrode” (when the term “substantially” is usedherein, it is meant that the recited characteristic, parameter, or valueneed not be achieved exactly, but that deviations or variations,including, for example, tolerances, measurement error, measurementaccuracy limitations, and other factors known to those of skill in theart, may occur in amounts that do not preclude the effect thecharacteristic was intended to provide). Additional (external) gas-puffvalves are illustrated substantially at the same axial position radiallyopposite to the internal gas-puff valve and arranged to provide aseparate portion of the filling gas through openings in the “OuterElectrode” of the FuZE SFS Z-pinch experimental device.

The arrangement illustrated in FIG. 3 of Shumlak '20 depends uponneutral gas diffusion from the gas-puff valves' locations into thesurrounding evacuated volumes for creation of substantially axisymmetricneutral gas density profiles generally having maxima at the puff-valves'axial position. Such profiles may provide sufficient fuel gas to drivethe sheared velocity flow for a duration of time commensurate with aduration of the Z-pinch discharges. After exhaustion of a neutral gasinventory (for example, because of driven outflow around the Z-pinchplasma column and/or diffusion of gas into other regions of the enclosedvolume), the Z-pinch current may decay because of instabilities even ifat least a portion of energy from high-voltage power supplies is stillavailable. For purposes of augmenting and improving the sheared flowprofile created by neutral gas injection, injection of pre-ionized gasusing plasma injectors, plasma guns, or ion sources may be employed inconjunction.

Accordingly, in at least one embodiment, one or more valves (e.g., oneor more gas-puff valves and one or more plasma injectors) may be fluidlycoupled to a fuel gas supply and configured to direct sufficient fuelgas (e.g., the neutral gas and/or the pre-ionized gas) sourced from thefuel gas supply to drive the sheared velocity plasma flow for theduration of each of the Z-pinch discharges. Specifically, in one suchembodiment, sufficient neutral gas may be directed to support alocalized breakdown path between inner and outer electrodes and toestablish the sheared velocity plasma flow. In an additional oralternative embodiment, sufficient pre-ionized gas may be directed tomaintain the sheared velocity plasma flow (e.g., to replenish theneutral gas).

Pertinent to the methods and devices described herein, the shearedvelocity flow stabilization is supported at least by the followingmodeling results. The axisymmetric plasma configuration representativeof the FuZE SFS Z-pinch experimental device has been simulated using theWARPXM computer code [U. Shumlak, R. Lilly, N. Reddell, E. Sousa, and B.Srinivasan, “Advanced physics calculations using a multi fluid plasmamodel.” Comput. Phys. Comm. 182, 1767 (2011)] based upon a nonlinearfive-moment, two-fluid (5M2F) plasma model. This model includesBraginskii-based viscosity and thermal conduction effects.

Some selected simulation results are illustrated in FIGS. 1-5 . Theresults for no sheared velocity flow are illustrated for reference inFIG. 1 . The results for three cases with three different parabolicsheared flow velocity profiles are illustrated in FIGS. 2-4 . The casesmay be distinguished by different sheared flow velocity values “vsfa” atr=a (where a is the nominal radius of the Z-pinch plasma) normalized bythe characteristic Alfvén speed v_(A) at the pinch edge (r=a). Initialconditions (t=0) for the illustrated cases include Bennett equilibriawith peak densities of 4×10²⁴ m⁻³, ion and electron temperatures of 1.27keV, and peak magnetic fields of 33.0 T. The effective pinch radius (a)is 0.91 mm.

The simulations use a specific normalized diffusivity limit (difflim=32m²·s⁻¹) and impose a minimum diffusivity (diffmin=3.2 m²·s⁻¹). Theelectrons are given viscosity with a diffusivity level equal to theminimum diffusivity. A perturbation is used in each case to trigger amode with wavelength equal to the axial length of the domain.

In FIG. 1 , vsfa=0.0 indicates the reference case having no shearvelocity flow. Snapshots of two-dimensional (r/z) cross sections 100 and110 of normalized ion densities at time points t=0.000 (corresponding toestablishment of the unperturbed Z-pinch plasma column) and t=8.000(normalized to the radial Alfvén time τ_(A)), respectively, areillustrated in grayscale. In addition, axial velocity (v_(z)) profiles120 and 130 (normalized by the characteristic Alfvén speed v_(A)) havesubstantially no shear, as illustrated in the v_(z)/v_(A) versus r/agraphs of FIG. 1 .

The case represented in FIG. 1 supports an understanding that a Z pinchwith no stabilization exhibits fast growing instabilities, for example,indicated by density perturbations 150 exhibiting substantial plasma ionlosses growing in a relatively small number of Alfvén time scale unitsmeasured by the time needed for a magnetized plasma perturbation topropagate from the axis (r=0) to the edge (r=a) of the unperturbedplasma column.

Three additional modeling results of a sheared velocity flow stabilizedZ pinch are illustrated in FIGS. 2-4 . The case illustrated in FIG. 2 ischaracterized by vsfa=0.25. Snapshots of two-dimensional (r/z) crosssections 200 and 210 of normalized ion densities at time points t=0.000(corresponding to establishment of the unperturbed Z-pinch plasmacolumn) and t=14.000 (normalized to τ_(A)), respectively, areillustrated in grayscale. An initial (t=0) parabolic sheared velocityprofile 220 evolves into profile 230 at the (normalized) time t=14,still exhibiting substantial shear outside the initial plasma columnboundary r=a. Ion density perturbations 250, although detectable, arepredominantly localized at radii commensurate to the initial r=a radiusof the unperturbed Z-pinch plasma column.

The case illustrated in FIG. 3 , characterized by vsfa=0.5, illustratesstronger stabilization effects relative to the cases illustrated byFIGS. 1 and 2 . Snapshots of two-dimensional (r/z) cross sections 300and 310 of normalized ion densities at time points t=0.000(corresponding to establishment of the unperturbed Z-pinch plasmacolumn) and t=26.400 (normalized to τ_(A)), respectively, areillustrated in grayscale. An initial (t=0) parabolic sheared velocityprofile 320 evolves into a perturbed ion axial velocity profile 330 atthe (normalized) time t=26.4, exhibiting nearly parabolic radialdependency and substantial shear outside the initial plasma columnboundary r/a=1. Ion density perturbations 350 are predominantlylocalized to a volume inside the plasma column (r<a).

The case illustrated in FIG. 4 is characterized by vsfa=0.75. Theresults in FIG. 4 illustrate stronger sheared-flow stabilization effectsrelative to the cases illustrated by FIGS. 1-3 . Snapshots oftwo-dimensional (r/z) cross sections 400 and 410 of normalized iondensities at time points t=0.000 (corresponding to establishment of theunperturbed Z-pinch plasma column) and t=37.000 (normalized to τ_(A)),respectively, are illustrated in grayscale. An initial (t=0) parabolicion axial sheared velocity profile 420 evolves into a (slightly)perturbed ion axial velocity profile 430 at the (normalized) time t=37,exhibiting substantially parabolic radial dependency. Ion densityperturbations 450 are predominantly localized to a volume inside theplasma column (r<a).

Some consequences of the WARPXM computer code are illustrated in FIG. 5. The graphic depicted in FIG. 5 illustrates time dependence of theintegrated Z-pinch mass over t/τ_(A) (a normalized time corresponding tothe normalized time t in FIGS. 1-4 ). The dependence in FIG. 5emphasizes effects of the sheared flow velocity values characterized bythe “vsfa” values. It may be observed that, for the case having nostabilization (vsfa=0.0), while the confinement of the Z-pinch plasmastarts to degrade after t=5 (marked by an initial decay of normalizedmass 500) and exhibits significant loss by nominal time t=8.0, thecorresponding mass ratios 525, 550, and 575 (vsfa=0.25, 0.5, and 0.75,respectively) exhibit increasingly enhanced plasma confinement,indicating that stabilized Z-pinch plasma may be sustained as long assufficient axial plasma current is supplied, and sufficient sheared,azimuthally symmetric ion velocity flow surrounding the plasma column isgenerated and maintained.

At least in context of the above, embodiments of devices and methods forgeneration and maintaining of azimuthally symmetric sheared ion velocityflow in accordance with the present disclosure are recited below. Somecomponents of a particular embodiment of a plasma (confinement) devicefor a stabilized Z pinch are illustrated in FIGS. 6A and 6B in anisometric view 600 (FIG. 6A) and a cross-sectional view 610 (FIG. 6B).

In general, the Z-pinch plasma device having a vacuum vessel asillustrated in FIGS. 6A and 6B (with associated systems, such aselectrical cables and conduits, vacuum pumps and ducts, diagnosticfeedthroughs, optical windows, and the like, omitted for clarity) may beenlarged relative to certain other Z-pinch plasma devices, at leastexcepting the neutral gas feeding valves (discussed in detail below)pertinent to the processes of improved sheared azimuthal velocity flowgeneration and maintenance as per certain embodiments provided in thepresent disclosure.

More particularly, in at least one embodiment, an acceleration volume620 may be increased relative to that of certain other Z-pinch plasmadevices and arranged to be filled by a gas mixture (e.g., a neutralworking gas mixture) via at least one internal valve 630, such as atleast one gas-puff valve (to provide neutral gas to the accelerationvolume 620) and/or plasma injector 630 (to provide pre-ionized gas tothe acceleration volume 620), arranged substantially along a centralaxis of the acceleration volume 620. Additionally or alternatively, aplurality of external valves, such as a plurality of gas-puff valves (toprovide neutral gas to the acceleration volume 620) and/or plasmainjectors 640 (to provide pre-ionized gas to the acceleration volume620), may be installed as a regular array on an external vacuum boundarywhich may be arranged as an external, or outer, electrode 650.

Depending upon the particular embodiments, the gas-puff valves and/orplasma injectors 630, 640 may be electronically triggered to deliver a“puff” of filling neutral and/or pre-ionized gas to start at a startingtime programmable down to a fraction of a ms and have a duration up toseveral hundred μs (e.g., up to 1 ms). An amount of filling gas (alsoreferred to herein as “fuel gas”) delivered (e.g., in the “puff”) mayalso be controlled by adjustments of a filling gas pressure supplied tothe gas-puff valves and/or plasma injectors 630, 640, eitherindividually or as a chosen subset of valves (where a subset of valvesmay include only a portion of valves and/or injectors 630, 640 or allvalves and/or injectors 630, 640). In addition, different valves and/orinjectors 630, 640 (or different combinations of multiple valves and/orinjectors 630, 640) may be fed by different fill gas mixtures having,for example different elemental ratios of filling gases and/or differentisotopic ratios (e.g., adjustable D₂/T₂ molecular ratios). In at leastone embodiment, the various gas-puff valves and/or plasma injectors maybe uniform (e.g., all of the same type/size with all the sameoperational settings, if so configurable), although in otherembodiments, different valves may be used for different locations. Inadditional or alternative embodiments, one or more gas-puff or other gasvalves and/or plasma injectors may control a flow of gas into theacceleration volume 620 via a manifold including multiple portsproviding passage into the acceleration volume 620. In such embodiments,the ports of the manifold may be uniform or may vary in configuration(e.g., to deliver different amounts of gas to different locations of theacceleration volume 620 when a respective valve is open).

Similar to neutral gas injection via gas-puff valves, ionized gas orplasma may be injected using combinations or manifolds of variouslylocated plasma injectors. Plasmas formed from gas mixtures may also becreated and injected in a manner similar to neutral gas injection.Plasma injection may provide a finer control of an eventual axial plasmadistribution as well as a shear flow profile thereof, which in turn mayallow for higher fidelity control of plasma stability and lifetime.Additional control of plasma injection may be provided due to the plasmaparticles being charged particles that may be accelerated by electricfields created by a variable electrical bias (or voltage) on injectionelectrodes. Thus, a speed of the injected plasma may be finelycontrolled to allow for fine adjustment and optimization of breakdown ofany neutral gas present (e.g., in the acceleration volume 620).Moreover, the injected plasma may travel at faster velocities thaninjected neutral gas, which may travel in a nearly static fashion(relative to the injected plasma) during Z-pinch discharge pulses. Assuch, relative to neutral gas injection, plasma injection may providepre-ionized fuel “on demand” (e.g., more immediately), for example, toreplenish the fuel gas during Z-pinch discharge pulses.

In some embodiments, plasma to be injected into the acceleration volume620 may be generated by pre-ionizing neutral gas with a spark plug orvia inductive ionization. More broadly, the gas-puff valves and/orplasma injectors 630, 640 may include one or more electrode plasmainjectors and/or one or more electrodeless plasma injectors. In exampleswherein the one or more electrode plasma injectors are included, theplasma to be injected into the acceleration volume 620 may be generated,at least in part, by electrode discharge. In additional or alternativeexamples wherein the one or more electrodeless plasma injectors areincluded, the plasma to be injected into the acceleration volume 620 maybe generated, at least in part, by inductive discharge produced by anexternal coil window (e.g., a radio-frequency antenna operating at 400kHz, 13.56 MHz, 2.45 GHz, and/or other frequencies permitted for use ina given local jurisdiction, e.g., within frequency ranges permitted bythe Federal Communications Commission). In some embodiments, neutral gasfor pre-ionization may be limited by a configuration of a neutral gasreservoir (not shown in FIGS. 6A and 6B) and/or neutral gas conductanceto a selected plasma injector configuration.

In some embodiments, axial distribution of the injected plasma may beensured via an axisymmetric plasma injector configuration. In at leastone embodiment, eight plasma injectors 640 may be respectivelypositioned at eight equally spaced ports of the manifold. The eightports may each be configured at an oblique angle (e.g., between 5° and90° with respect to the central axis of the acceleration volume 620)with respect to a housing of the acceleration volume 620. In oneexample, the oblique angle may be 45° with respect to the central axisof the acceleration volume 620. In some embodiments, the eight ports maybe configured at a single axial position along the central axis of theacceleration volume 620 (that is, the eight ports may be equally spacedabout a circumference or other perimeter of the acceleration volume 620at the axial position). In other embodiments, the ports may includemultiple sets of eight ports, with each set of eight ports being equallyspaced about a different axial position along the central axis of theacceleration volume 620. In an example embodiment, the sets of eightports may be configured as interleaved pairs of sets, wherein a firstset of eight ports may be positioned at a first axial location and asecond set of eight ports may be positioned at a second, different axiallocation and rotated relative to the first set such that each port ofthe second set is positioned between a pair of ports of the first setwith respect to the circumference of the acceleration volume 620.Specifically, in such an embodiment, each port of the first set of eightports may be spaced around the circumference of the acceleration volume620 every 45°, and each port of the second set of eight ports may bespaced around the circumference of the acceleration volume 620 every 45°offset (rotated) from the first set of ports by 22.5°, such that oneport of the first and second sets is provided around the circumferenceof the acceleration volume 620 every 22.5°. In additional or alternativeembodiments, plasma injection may be performed azimuthally, e.g., alonga chord perpendicular to the central axis of the acceleration volume620, so as to generate an azimuthal flow within the acceleration volume620. In additional or alternative embodiments, the valves may beconfigured differently (e.g., asymmetrically distributed azimuthallyand/or with different angular distributions) with other variations toachieve a substantially equivalent profile by compensating for effectsof the variations.

In some embodiments, injecting the acceleration volume 620 withpre-ionized gas may result in plasmas having a plasma temperature in arange of 1 to 10 eV. Moreover, and as noted above, because an injectionvelocity of pre-ionized gas may be significantly greater than that ofneutral gas, a velocity of the plasma within the acceleration volume 620may be up to 50×10³ m/s. In some embodiments, injection of pre-ionizedgas may provide flexibility in an amount of particles injected.Specifically, in an example embodiment, an amount of pre-ionized gasparticles may be injected in 1/50 of a time utilized to inject the sameamount of neutral gas particles. For example, an amount of time utilizedto inject 10 Torr-L of neutral gas particles (where 1 Torr-L isproportional to 2.5×10¹⁹ molecules at 273 K) may be the same amount oftime utilized to inject 500 Torr-L of pre-ionized gas particles.Similarly, in some embodiments, an injection rate (or mass flow rate) ofpre-ionized gas may be varied according to power supply current andvoltage (that is, a waveform of an injection pulse). As an example,increasing the power supply voltage (e.g., to between 100 V and 500 V)may concomitantly increase the injection velocity. As another example,increasing the power supply current (e.g., to between 1 A and 500 A) mayconcomitantly increase the injection rate.

Injection of neutral gas in particular may be accomplished through puffvalves or through release of hydrogen gas from a metal hydride, e.g.,titanium deuteride (TiD₂) or other metal hydrides based on scandium,vanadium, or other metals. In some embodiments, the puff valves may besolenoid-driven puff valves (though other configurations may beimplemented and are within the scope of the present disclosure).

As discussed above, the at least one internal gas-puff valve and/orplasma injector 630 and the plurality of external gas-puff valves and/orplasma injectors 640 may be activated either individually or in groups.An initial gas load inside the acceleration volume 620 having desiredaxial and azimuthal profiles may be achieved by timing individual valvesor groups of valves. Such valves (or groups thereof) may be timed in afashion to align an arrival of the neutral and/or pre-ionized gas and/ormixtures thereof to a desired initial profile, such as the embodimentsdiscussed in detail below and illustrated in FIGS. 8 and 10A-10F. Powersupplies (not shown at FIGS. 6A and 6B) may be timed to achieveionization at a desired axial location and utilize the initial gas loadto produce and sustain the sheared flow.

Various combinations of (neutral gas) gas-puff valves with plasmainjectors may be activated to further adjust (e.g., optimize) suchparameters, e.g., to achieve a desired level of power output. Moreover,plasma may be injected into the acceleration volume 620 significantly(e.g., ˜100×) faster than puffed neutral gas. A combination of suchdifferent injection speeds allowed by acceleration of plasma injectionwith neutral gas injection provides an even larger parameter space foroptimization. Additionally, plasma injectors may serve to inject massand carefully control locations of neutral gas ionization.

The embodiment illustrated in FIGS. 6A and 6B incorporates theacceleration volume 620 prefabricated so as to incorporate connectors orother coupling elements for the at least one internal gas-puff valveand/or plasma injector 630 extending from within an internal, or inner,electrode 660. For example, the at least one internal gas-puff valveand/or plasma injector 630 [and respective coupling element(s)] mayinclude eight valves 630 symmetrically distributed azimuthally (having45° angular separation) at z=−50 cm (relative to a z=0 position at anunsupported end 665 of the inner electrode 660, where the z-axis iscoincident with the central axis of the acceleration volume 620 andwhere a negative direction of the z-axis extends from the unsupportedend 665 along a central axis of the inner electrode 660 and a positivedirection of the z-axis extends from the unsupported end 665 through theacceleration volume 620 in an opposite direction from the negativedirection), eight valves 630 similarly distributed at z=−75 cm, andeight valves 630 similarly distributed at z=−100 cm. The illustratedembodiment may be readily updated with additional valves to allow forinjection of more fuel gas (e.g., for longer lasting pinch discharges)and control of an axial pressure distribution of the filling neutral gasin the acceleration volume 620 (e.g., for additional enhancement of thesheared velocity shear flow duration). In additional or alternativeembodiments, the valves may be configured differently (e.g.,asymmetrically distributed azimuthally and/or with different angulardistributions) with other variations to achieve a substantiallyequivalent profile by compensating for effects of the variations. Suchconsiderations may apply equally to plasma injectors.

The gas-puff valves of the illustrated embodiment incorporate prismaticstructural elements which may allow tool effectors (e.g.,force-transmitting elements of hand tools or coupling inserts forpowered appliances) to be used therewith, so as to transfer torque andother associated stresses directly to more robust principal structuralelements while avoiding various inserts, connectors, contacts, vacuum orpressure sills, pottings, and/or solder joints.

Gas-puff valves of the present disclosure, in an embodiment, may bedesigned to incorporate an orifice diameter of no less than 0.075 in anda plenum volume of at least 1 cm³. In addition, one feature of thegas-puff valves in accordance with embodiments of the present disclosureis the capability to close (and stay closed) at a preprogramed timebefore and during the Z-pinch discharge.

As recited above (including the documents incorporated by reference),preservation of azimuthal symmetries of plasmas and associated shearedvelocity flow is one advantage of embodiments of the present disclosure.Accordingly, reproducible formation and shaping of initial azimuthallysymmetric plasma structures at predetermined axial positions in theacceleration volume 620 may be achieved by at least some embodiments ofthe present disclosure. In various embodiments, a variety of “plasmaformation” devices and methods may be used. Such devices and method mayinclude (but are not limited to): dedicated systems for plasmageneration [some of which may be configured to account for complexitiesassociated with specific power supplies and conduits and/or complex(pre)ionization subsystems], plasma injectors, and tuned operationalalgorithms and other methods.

The plasma confinement device of FIGS. 6A and 6B may include acontroller or other computing device (not shown), which may includenon-transitory memory on which executable instructions may be stored.The executable instructions may be executed by one or more processors ofthe controller to perform various functionalities of the plasmaconfinement device. Accordingly, the executable instructions may includevarious routines for operation, maintenance, and testing of the plasmaconfinement device. The controller may further include a user interfaceat which an operator of the plasma confinement device may enter commandsor otherwise modify operation of the plasma confinement device. The userinterface may include various components for facilitating operator useof the plasma confinement device and for receiving operator inputs(e.g., requests to generate plasmas for thermonuclear fusion, etc.),such as one or more displays, input devices (e.g., keyboards,touchscreens, computer mice, depressible buttons, mechanical switchesother mechanical actuators, etc.), lights, etc. The controller may becommunicably coupled to various components (e.g., valves, powersupplies, etc.) of the plasma confinement device to command actuationand use thereof (wired and/or wireless communication paths between thecontroller and the various components are omitted from FIGS. 6A and 6Bfor clarity).

Some aspects of plasma initiating and subsequent shaping parts pertinentto embodiments of the present disclosure are schematically illustratedin FIG. 7 . These shaping parts may include sharp points (e.g., tipsformed at localized concave elements 725; see below) that enhance alocal electric field and assist in plasma breakdown. Variousconfigurations of “passive” (that is, without dedicated power or gassupplies that actively drive field emission) shaping parts 700 may bearranged in a form of a ring electrode fit in one or more recesses inthe inner electrode 660 at one or more negative z axial positionsgenerally in proximity of one or more internal gas-puff valves 630(inner electrode 660 and gas-puff valves 630 not shown at FIG. 7 ; seeFIGS. 6B and 8 ). One significant functionality of such parts, in atleast one embodiment, is to initiate and sustain a multichannelbreakdown of surrounding neutral gases (starting with varioussubstantially independent azimuthally distributed radial dischargestreamers) conducive to creation and sustainment of substantially equalcurrent flow in all radial directions. Note that the shaping part 700may be placed at a location of the gas valves or downstream therefrom(e.g., to the right in FIGS. 6B and 8-10F) in various embodiments.Though the sharp shaping parts are described with reference to the innerelectrode 660, such features may be used at either the inner electrode660, the outer electrode 650 (not shown at FIG. 7 ; see FIGS. 6A, 6B,and 8 ), or both. In some embodiments, enhancing the electric field nearsuch sharp points is beneficial at the cathode (from which electrons areemitted), which might be either the inner electrode 660 or the outerelectrode 650. Sharp points on a surface of the anode may also beincluded such that a breakdown path may be selectively establishedbetween the shaping parts on the cathode and the shaping parts on theanode (which may be, for example, the inner electrode 660 and the outerelectrode 650, respectively).

The shaping part 700, illustrated in FIG. 7 , incorporates a conductivering 710 arranged to include at least one contact surface 720 forming alow contact resistance surface contact (e.g., to cause a voltage dropbetween the at least one contact surface 720 and the inner electrode 660of less than 100 V) with a cylindrical or conical or otherwise taperedouter surface of the inner electrode 660. In some embodiments, theconductive ring 710 may be formed from one or more conductive materialswhich may be fully or at least partially chemically and/orthermo-mechanically compatible (e.g., the heat and stresses experiencedduring operation will not appreciably impact lifetime use of theconductive ring 710) with conductors of the inner electrode 660. Inaddition, a plasma-facing portion 715 of the shaping part 700 may beformed from conductors resistant to chemical and physical damage bysupported discharges. In many embodiments, one or more refractory metals(such as one or more of W, Ta, Nb, Mo, or Re; additionally oralternatively including one or more of Ti, V, Cr, Mn, Zr, Tc, Ru, Rh,Hf, Os, or Ir) and/or alloys or combinations thereof may be utilized atleast for relatively low chemical reactivities, relatively high meltingpoints, and relatively high resistance to plasma ablation andsputtering.

In additional or alternative embodiments, the plasma-facing materialsmay be based on conductive forms of carbon including graphite, sinteredor pressed carbon powders, carbon fiber matrices, and/or carbon nanotubeincorporating structures and compositions. In addition to relativeinsensitivities to plasma effects, degradation, and damages,carbon-based structures (especially carbon nanotubes) may exhibitelectron multiplication properties desirable during plasma generationphases.

In other embodiments, plasma-facing portions 715 may be textured toincorporate a multitude of localized concave elements 725 so as to formstructured arrays. Such elements, in certain embodiments, may enhancelocal electric fields and facilitate electron field emission from solid(and liquid) surfaces. The elements 725 may be formed by mechanicalaction (including cutting, scratching, sanding, sandblasting, grooving,checkering, stumping, embossing, knurling, etc.). Also differentchemical and/or thermal processes (e.g., etching, chemical deposition,spraying, sputtering, ion and neutral implantation, epitaxial growth,etc.) may also be involved. In certain embodiments, multitudes of theelements 725 having relatively small characteristic dimensions (e.g., incomparison with a size of the plasma-facing portions 715) may beproduced and maintained to avoid significant changes in geometry when/ifany individual elements 725 get damaged or deformed (e.g., by arcing orlocal overheating). For example, the elements 725 may have an averageheight of 1-10% of a height of the plasma-facing portions 715 (excludingthe elements 725), in some embodiments.

In at least one embodiment, the shaping part 700 (whether including orexcluding the elements 725) around the inner electrode 660 may beconfigured as a generally uniform annular ring, varying, for example,where ports for the internal gas-puff valves 630 traverse a crosssection of the plasma-facing portions 715 and the at least one contactsurface 720. However, in some embodiments (e.g., such as when theshaping part 700 is configured as the generally uniform annular ring),the shaping part 700 may have a varied cross section around and/or alongthe inner electrode 660. Other variations, such as where multiplediscrete shaping parts 700 that do not form a single ring are placedaround the inner electrode 660, are also within the scope of the presentdisclosure.

Also, as discussed in detail below with reference to FIG. 8 , a shapingpart similar to the shaping part 700 shown in FIG. 7 may form a ringalong an interior surface of the outer electrode 650. Such a shapingpart and corresponding contact surface (similar to the at least onecontact surface 720) may be configured such as shown in FIG. 7 (oraccording to variations discussed herein), except with the cross sectionof the at least one contact surface 720 and the plasma-facing portion715 being rotated 180° to accommodate for being on the interior surfaceof the outer electrode 650 instead of an exterior surface thereof. Aslope of the at least one contact surface 720 may also be different thanillustrated in FIG. 7 to accommodate tapering or other surfacevariations of the outer electrode 650 or lack thereof (see, e.g.,discussion of the “tapered electrodes configuration” below in referenceto FIG. 9 ).

In addition to the shaping part 700 illustrated in FIG. 7 , othermethods of assisting/controlling/directing plasma formation may be usedindividually or in combination. One possibility includes usingradioactive material(s) embedded in the outer electrode 650 and/or innerelectrode 660. Specifically, high-energy particles or photons emittedfrom radioactive decay processes may result in pre-ionization near theembedded radioactive material(s), resulting in a region having increasedplasma breakdown relative to regions not including the embeddedradioactive material(s). For example, beta emitters and/or gammaemitters may be selected for the embedded radioactive material(s).

An additional or alternative possibility includes illuminating the outerelectrode 650 and/or the inner electrode 660 with incident laser lightat a region where plasma pre-ionization is desired. Within such regions,electrode surfaces (e.g., of the outer electrode 650 and/or the innerelectrode 660) may include materials chosen specifically to emit X-rayor other forms of ionizing radiation when subjected to the incidentlaser light. An additional or alternative possibility includes usinglaser light to directly ionize gas (e.g., via direct interactions of thelaser light with neutral gas particles). In such embodiments, the laserlight may pass through the neutral gas, and deposit energy throughoutthe laser path, resulting in pre-ionization and a directed channel forplasma breakdown (e.g., a channel having greater plasma breakdown than asurrounding volume).

Other methods to assist/control/direct plasma formation may utilizevarious forms of cathodes, such as field emitters or thermionicemitters, located on the electrode surfaces (e.g., of the outerelectrode 650 and/or the inner electrode 660) where breakdown isdesired. Field emitters may use relatively high electric fields to emitelectrons from small sharp features. Examples of such emitters mayinclude nanostructures such as carbon nanotubes, graphene emitters,nanowire emitters, Schottky emitters, etc. Additionally oralternatively, thermionic emitters may be used to cause plasmabreakdown. Examples of such emitters may include heated tungstenfilaments which emit electrons at relatively high temperature. Schottkyemitters may be considered field enhanced thermionic emitters.

One embodiment of a process of initiating and driving the azimuthallysymmetric sheared flow for stabilization of Z-pinch discharges in aplasma confinement device, such as the plasma confinement devicedescribed in detail above with reference to FIGS. 6A-7 , isschematically illustrated in FIG. 8 . The process, in this example, maybe characterized by the schematic illustrations of steps or stages 810,820, 830, 840, 850, 860, 870, and 880 of generally unequal duration. Incertain embodiments, the process may include executing steps 810, 820,830, 840, 850, 860, 870, and 880 in sequence.

In some embodiments, the process, or a portion thereof, may beimplemented as executable instructions stored in non-transitory memoryof a computing device, such as a controller communicably coupled to theplasma confinement device. Moreover, in certain embodiments, additionalor alternative sequences of steps may be implemented as executableinstructions on such a computing device, where individual stepsdiscussed with reference to the process may be added, removed,substituted, modified or interchanged.

The process begins with the step 810, which may include application of ahigh-voltage generating radial electric field (not shown) between theelectrodes 650 and 660, and sequential activation of one or moreinternal and external valves 630 and 640 (in possible combination withplasma injectors 640). The valves 630 and 640 may be arranged to locallyintroduce initial measured and predetermined concentrations 812 offilling gas. In certain embodiments, it may be desirable to initiate gaspuffing and/or plasma injection during an initialization phase andcontinue to deliver sufficient initial concentrations 812 of filling gasin a proximity of the shaping part 700 for additional protection againstpremature and/or asymmetric gas breakdowns.

During the step 820, the initial concentrations 812 may spontaneouslyevolve by neutral gas diffusion processes to form a continuous (e.g.,uninterrupted) axisymmetric volume of neutral filling gas formation 822occupying a substantial portion (e.g., a majority) of the accelerationvolume 620. In some embodiments, the volume of neutral filling gasformation 822 exhibits a neutral gas molecule number density gradient inan axial direction along the central axis of the acceleration volume 620(e.g., toward the unsupported end 665 of the inner electrode 660), whilesubstantially maintaining the azimuthal symmetry favorable forsubstantially symmetric distribution of discharge streamers duringinitial breakdown of the filling gas. In certain embodiments, theneutral gas molecule number density gradient can be such that Paschenbreakdown occurs at the shaping part 700. In additional or alternativeembodiments, pre-ionized gas may be injected at the shaping part 700 tofacilitate formation of an ionization wave that moves into the neutralgas that has been injected upstream (e.g., towards a supported end ofthe inner electrode 660 opposite the unsupported end 665).

During the step 830, a proximal electric field structure shaped by thegeometry and material properties of the shaping part 700 may facilitateneutral gas breakdown forming an axisymmetric plasma structure 835supporting current flow 837 between the inner electrode 660 and the(surrounding) outer electrode 650, e.g., localized axially in a vicinityof the shaping part 700. The current flow 837, supported by energy fromthe power supplies (e.g., capacitor banks or similar), may form acontinuous (e.g., uninterrupted) current loop (from the outer electrode650, through the plasma structure 835, and into and through the innerelectrode 660) which may generate a substantially azimuthal magneticfield 838 (as indicated by azimuthal field lines). In additional oralternative embodiments, to form the plasma structure 835, pre-ionizedgas may be injected from plasma injectors 640 towards the unsupportedend 665 of the inner electrode 660.

Lorentz force interactions between the current flow 837 and the magneticfield 838 may cause migration of the current flow 837 from the shapingpart 700 in the direction of the unsupported end 665, as illustrated inthe schematic representation of the step 840. In addition, the Lorentzforce interaction may induce the current flow 837 along the surface ofthe outer electrode 650.

During the step 850, the current flow 837 may continue to developtowards and up to the unsupported end 665. Simultaneously, a magneticpressure driven by the magnetic field 838 enclosed by the current flow837 may displace the developing plasma structure 835 in a direction ofthe opposing portion 655 of the outer electrode 650 arranged to face theunsupported end 665. In embodiments where present, the ionization wavethat moves into the neutral gas may be controlled by injecting varyingamounts of pre-ionized gas, e.g., at the internal valves 630 and/or theexternal valves 640. In at least one embodiment, substantially azimuthalsymmetry of the plasma structure 835 supporting the current flow 837 maycontribute significantly to an efficiency of the process, as anysignificant disturbances in the current flow 837 may causeinstabilities, electrode damage, and/or introduction of metallicimpurities into the developing discharge.

In certain embodiments, the discharge developing steps 830, 840, and 850may last from a fraction of a microsecond to a few microseconds, e.g.,significantly shorter than either the step 810 (corresponding to fillingthe acceleration volume 620 with neutral gas) or the steps 860, 870, and880 (corresponding to the Z-pinch discharge). As such, the neutralfilling gas formation 822 is illustrated as stationary, as the neutralfilling gas formation 822 may evolve over (e.g., only over) timeintervals significantly longer than a duration of the steps 830, 840,and 850.

In certain embodiments, plasma injection may occur between steps 810 and820. In additional or alternative embodiments, plasma injection mayoccur rapidly and on the same scale as steps 830, 840, and 850, and maybe used to control formation/initialization and dynamics of such steps.

The step 860 corresponds to initial Z-pinch operation step, includingformation of a Z-pinch plasma column 865 created to support a Z-pinchcurrent I_(pinch). In addition, a residual plasma structure 866 may beformed to support a residual (radial) current 867 flowing through theneutral filling gas formation 822 in the acceleration volume 620.Moreover, in various embodiments, propagation of the plasma structure835 may drive an initial sheared velocity plasma flow 868 surrounding(and stabilizing) the Z-pinch plasma column 865. In some embodiments,the residual plasma structure 866 may be initiated in the proximity ofthe shaping part 700 as characterized by a (locally) highest numberdensity of neutral gas constituents (molecules and/or atoms).

As discussed above, in at least one embodiment, during dischargesustainment in the step 870, the Z-pinch plasma column 865 may besustained and stabilized by continued plasma flow from the accelerationvolume 620. An ionization front 872 may continually produce plasmaaccelerated from the acceleration volume 620 by the residual current 867to drive the sheared velocity plasma flow 868.

During the step 880, in at least one embodiment, the ionization front872 may move towards a breech end of the acceleration volume 620,ionizing remaining neutral fuel gas in a continuous (e.g.,uninterrupted) manner, until all or substantially all fuel gas availablein the acceleration volume 620 is ionized, resulting in extinction ofthe ionization front 872 and subsequent disintegration of the Z-pinchcurrent I_(pinch) (e.g., a current through the Z-pinch plasma column 865and the inner electrode 660, such as the current 950 illustrated in FIG.9 , described below). When steps 810, 820, 830, 840, 850, 860, 870, and880 are completed, the plasma confinement system may be flushed toremove fusion byproducts and the process described above may be repeatedfor another pulse. In certain embodiments, the process and repetitionthereof may be automated and controlled by a software application, e.g.,implemented by the controller communicably coupled to the plasmaconfinement system.

Another embodiment of a plasma confinement device, a Z-pinch plasmadevice 900, is illustrated schematically in FIG. 9 . The Z-pinch plasmadevice 900 may generate a plasma within an assembly volume 635 of aplasma confinement chamber 615, the plasma confined, compressed, andsustained by an axially symmetric magnetic field. The axially symmetricmagnetic field may be stabilized by a sheared ion velocity flow drivenby electrical discharge between a pair of electrodes interfacing withthe plasma confinement chamber 615.

Devices belonging to the illustrated class of plasma confinement devicesare generally related with the previous embodiments discussed above andillustrated in FIGS. 6A-8 and have similar features excepting additionalor alternative subsystems and functionalities below. Excepting certainassembly and operational aspects which may arise from such differences,the description provided above with reference to FIGS. 6A-8 may beadditionally applied to the embodiment depicted in FIG. 9 . In certainembodiments, additional subsystems and/or functionalities may also beincluded in the Z-pinch plasma device 900 which were not described indetail above with reference to FIGS. 6A-8 and which may be additionallyapplied to the embodiments depicted in FIGS. 6A-8 .

In an example embodiment, the Z-pinch plasma device 900 may include anouter electrode 650 separated physically and functionally from anexternal vacuum boundary 910, the external vacuum boundary 910, togetherwith portions of the inner electrode 660, forming a vacuum vessel 645 asa low pressure container including the plasma confinement chamber 615.The intermediate electrode 920 may be positioned so as to have a radiusin between a radius of the inner electrode 660 and a radius of the outerelectrode 650. Specifically, the intermediate electrode 920 maysubstantially surround the inner electrode 660 and the outer electrode650 may substantially surround the intermediate electrode 920. Forexample, the inner electrode 660 may include one end 665 that is atleast partially surrounded by the intermediate electrode 920 and theintermediate electrode 920 may include one end 965 that is at leastpartially surrounded by the outer electrode 650.

The Z-pinch plasma device 900 may incorporate at least two functionallyseparate power supplies, e.g., at least one primary power supply 930primarily arranged and controlled to drive a Z-pinch (discharge) current950 (I_(pinch)), and at least one additional power supply 940 primarilyarranged and controlled to drive the residual current 867. In someembodiments, the at least one primary power supply 930 may be separatepower supply device(s) from the at least one additional power supply940. In other embodiments, the at least one primary power supply 930 andthe at least one additional power supply 940 may be components of thesame power supply device.

For example, in at least one embodiment, a single power supply devicemay have a plurality of outputs which individually provide an amount ofpower to enable performance of a respective function (e.g., drive theZ-pinch current 950, drive the residual current 867, etc.). Such anarrangement may be based on at least two power supplies (e.g., oneprimary power supply 930 and one additional power supply 940) and mayallow for additional control of the Z-pinch current 950 and sheared flowstabilization thereof. In principle, the at least two power supplies maybe scaled, charged, and controlled such that the Z-pinch current 950 andthe stabilization thereof may be maintained for commensurate timeperiods before any of the at least two power supplies prematurely runsshort or out of stored energy.

In certain embodiments, the Z-pinch plasma device 900 may incorporate a“tapered electrodes” configuration, characterized by broadening a gapbetween the inner electrode 660 and the intermediate electrode 920 bytapering, along the central axis of the acceleration volume 620, the end965 of the intermediate electrode 920 outwards to increase a volume ofat least a portion of the acceleration volume 620, e.g., in a directionof the (unsupported) ends 665 and 965. In one example, the taper may bebetween 0° and 15° from the central axis of the acceleration volume 620.Such an arrangement may facilitate a transfer of momentum from plasmaheated by the residual current 867 to neutral gas, e.g., along thecentral axis, thereby creating and sustaining sheared flowstabilization. The momentum transfer may be described and modeled usingmethodology applicable to design/optimization of “de Laval nozzles” asknown in the field of jet propulsion.

While techniques described herein are discussed in connection withthermonuclear fusion and, for example, harnessing energy productiontherefrom, the techniques described herein can be used for otherpurposes, such as heat generation (e.g., for manufacturing utilizingrelatively high temperatures) and propulsion. For example, theembodiment of FIGS. 6A-8 or the embodiment of FIG. 9 may be modified atleast by removing the vacuum chamber 338 or the external vacuum boundary910, respectively, and introducing an opening in one end of the outerelectrode 650 to allow fusion products to escape (e.g., parallel to thecentral axis of the acceleration volume 620). In one embodiment, amagnetic nozzle (not shown at FIG. 9 ) is positioned downstream of theouter electrode 650, e.g., to the right of the outer electrode 650 in aplane of FIG. 9 , to collimate the plasma to reduce any exhaust plumedivergence.

The Z-pinch plasma device 900 may include a controller or othercomputing device 948, which may include non-transitory memory on whichexecutable instructions may be stored. The executable instructions maybe executed by one or more processors of the controller 948 to performvarious functionalities of the Z-pinch plasma device 900. Accordingly,the executable instructions may include various routines for operation,maintenance, and testing of the Z-pinch plasma device 900. Thecontroller 948 may further include a user interface at which an operatorof the Z-pinch plasma device 900 may enter commands or otherwise modifyoperation of the Z-pinch plasma device 900. The user interface mayinclude various components for facilitating operator use of the Z-pinchplasma device 900 and for receiving operator inputs (e.g., requests togenerate plasmas for thermonuclear fusion, etc.), such as one or moredisplays, input devices (e.g., keyboards, touchscreens, computer mice,depressible buttons, mechanical switches or other mechanical actuators,etc.), lights, etc. The controller 948 may be communicably coupled tovarious components (e.g., valves, power supplies, etc.) of the Z-pinchplasma device 900 to command actuation and use thereof (wired and/orwireless communication paths between the controller 948 and the variouscomponents are omitted from FIG. 9 for clarity).

FIGS. 10A-10F illustrate schematically an embodiment of a process ofinitiating and driving the azimuthally symmetric sheared flow forstabilization of Z-pinch discharges in a plasma confinement device. Theseries of FIGS. 10A-10F show two configurations, a first configurationwhere the inner electrode is a cathode and the outer electrode is ananode (on a left side of each of FIGS. 10A-10F when FIGS. 10A-10F areoriented so that alphanumeric characters depicted therein are orientedin standard fashion), and a second configuration where the innerelectrode is an anode and the outer electrode is a cathode (on a rightside of each of FIGS. 10A-10F when FIGS. 10A-10F are oriented so thatalphanumeric characters depicted therein are oriented in standardfashion). Certain Z-pinch plasma confinement devices may correspond tothe first configuration, which may be simpler to construct and operatesuccessfully. However, as shown in more detail in the discussion belowwith respect to FIGS. 11-24 , the second configuration may yieldadvantageous and unexpected results in accordance with variousembodiments described herein. Certain non-alphanumeric symbols used inFIGS. 10A-10F (e.g., current flow arrows, gas valves, gas clouds,magnetic field symbols, gas flow arrows, etc.) are the same as used inFIG. 8 . Note that portions labeled with “cathode” and “anode” may beelectrically connected to portions with the same name (“cathode” or“anode”). Note that while FIGS. 10A-10F show a set of gas valves inphysical contact with or directly adjacent to the outer electrode, otherconfigurations are also within the scope of the present disclosure, suchas the configurations of valves illustrated and described in detailabove with reference to FIG. 8 , which may or may not include one ormore shaping parts, such as described in detail above with reference toFIG. 7 .

In at least one embodiment, greater stability of Z-pinch plasma may beachieved with plasma confinement systems such as disclosed herein wherethe outer electrode is a cathode and the inner electrode is an anode.Specifically, and as discussed in greater detail below with reference toFIGS. 11-24 , the Z-pinch m=0 instability, and its stabilization byradially sheared axial flow, is studied using the nonlinear ideal 5M2Fmodel, and an extension of that model to include Braginskii heat andmomentum transport. Using the ideal 5M2F model, linear growth rateresults are compared with prior work using magnetohydrodynamics (MHD)and Hall MHD. In scenarios with and without radially sheared axial flow,agreement with Hall MHD is excellent, suggesting that among thetwo-fluid terms the Hall term is dominant. In the limit of small ioninertial length, results also match MHD. A comparison withparticle-in-cell (PIC) modeling of shear-free m=0 stability focuses on aplasma scenario based on recent experimental results. In a scan of modewavenumber, ideal 5M2F results are qualitatively similar to PIC: growthrate rises to a peak at moderate wavenumber, and decline at largewavenumber, in contrast to MHD results, which show saturation of growthrates with increasing wavenumber rather than a decline. The peaknormalized 5M2F growth rate is γτ_(A)=1.5, where τ_(A) is the Alfvéntransit time across the pinch. The peak occurs at normalized wavenumberka=10, where a is the effective pinch radius. For comparison, PICresults have peak growth of γτ_(A)=0.77 at ka=5. IncludingBraginskii-based closure of the 5M2F model does not qualitatively changethe ideal results in this particular case. Nonlinear 5M2F modeling withthe dissipative Braginskii-based closure is done in cases withpinch-edge sheared-flow speed equal to half the Alfvén speed. Nonlinearmixing due to sheared flow yields a saturated quasi-steady state withmodest losses of pinch ion inventory and pinch thermal energy:approximately 30% and 10%, respectively. 5M2F modeling capturesessential physics of m=0 instability and offers a computationallytractable route to high-fidelity modeling of 3D Z-pinch behavior,including m=1 instability.

Experimental evidence from SFS Z-pinch research, together with seminalnumerical stability analysis suggests that radially sheared axial flowenables the observed long plasma lifetimes; static Z pinches aretypically terminated by m=0 (sausage) and m=1 (kink) instabilities,which have growth rates near the radial Alfvén transit frequency.

Insight from computational modeling is expected to be a vital part ofongoing SFS Z-pinch development. As plasma parameters are increased infuture experiments, high-fidelity modeling can be used to explorevarious processes involved in plasma formation, assembly, andconfinement. Processes of particular interest include, for example, thedeflagration mode that is linked to sustainment of sheared floweffective resistivity due to electron drift microturbulence, and thedynamics of the flowing Z pinch itself.

The five-moment multi-fluid model is an excellent candidate foraccurately capturing the physics of interest. The five-moment two-fluid(5M2F) plasma model (with the two fluids representing ions andelectrons) has previously been applied to model Z-pinch instabilities,capturing realistic m=0 growth rates and an interesting electron driftinstability when the electron cross-field drift velocity exceeds the ionthermal speed. The 5M2F model may allow an effective compromise betweenthe fidelity (but prohibitive computational expense) of a kinetic modeland the computational tractability (but limited fidelity) of an approachbased on MHD.

Research presented herein applies the 5M2F model to explore m=0instabilities, with and without sheared flow. The 5M2F model includesfinite-inertial-length corrections for ions and electrons, and also theeffects of finite speed of light. In the limit that electron inertia isnegligible and light speed is infinite, results should be match Hall-MHDresults. In some examples, a linearized Hall-MHD model is applied tostudy m=0 stability of a Bennett equilibrium with a parabolicsheared-flow profile (v_(sf)∝r²). Additional work has been done usingthe same setup, but with a nonlinear MHD model, considering both linear(v_(sf)∝r) and parabolic sheared-flow profiles. Hereinbelow, the 5M2Fmodel is benchmarked against these MHD and Hall-MHD results.

To assess the physical fidelity of the 5M2F model with full Braginskiiclosure, comparison is made with fully kinetic (i.e., not gyrokinetic orotherwise reduced) PIC modeling of m=0 Z-pinch stability, includingCoulomb collisions. In that work, the PIC model is applied to study aFuZE-like Bennett profile with a linear sheared-flow profile. A scan ofka without sheared flow shows growth rates similar to MHD results up toka=5, at which point the PIC growth rate reaches a maximum. For largerk, the PIC results show decreasing growth rates, unlike MHD resultswhich typically show constant or increasing growth rates at large k.Simulations at ka=5 with sheared flow show m=0 stability for v_(sf)^(a)≥0.75. Stability analysis performed at reactor-like conditions, inwhich the collisionality is reduced, yields similar results.

Five-moment fluid equations for a given species are derived by takingmoments of the associated Boltzmann equation. As described byBraginskii, the first three moments of the Boltzmann equation forspecies α yield evolution equations for five independent variables:number density (n_(α)), three momentum components (m_(α)n_(α)v_(α),where m_(α) and v_(α) are the species mass and velocity), and a scalarpressure (p_(α)). The five-moment multi-fluid model, and its reductionto 5M2F, is summarized below, and Braginskii closure details arepresented below. Implementation of the model in the WARPXM framework isalso discussed below.

Moments of the Boltzmann equation for species a yield the followingfluid equations:

$\begin{matrix}{{{\frac{\partial n_{\alpha}}{\partial t} + {\nabla \cdot \left( {n_{\alpha}v_{\alpha}} \right)}} = S_{\alpha}^{n}},} & (1)\end{matrix}$ $\begin{matrix}{{{{\frac{\partial}{\partial t}\left( {m_{\alpha}n_{\alpha}v_{\alpha}} \right)} + {\nabla \cdot \left( {{m_{\alpha}n_{\alpha}v_{\alpha}v_{\alpha}} + {p_{\alpha}{\mathbb{I}}} + \prod\limits_{\alpha}} \right)}} = {{q_{\alpha}{n_{\alpha}\left( {E + {v_{\alpha} \times B}} \right)}} + S_{\alpha}^{m}}},} & (2)\end{matrix}$ $\begin{matrix}{{{\frac{\partial e_{\alpha}}{\partial t} + {\nabla \cdot \left\lbrack {{e_{\alpha}v_{\alpha}} + {v_{\alpha} \cdot \left( {{p_{\alpha}{\mathbb{I}}} + \prod\limits_{\alpha}} \right)} + h_{\alpha}} \right\rbrack}} = {{q_{\alpha}n_{\alpha}{v_{\alpha} \cdot E}} + S_{\alpha}^{e}}},} & (3)\end{matrix}$where e_(α)=m_(α)n_(α)v_(α) ²/2+p_(α)/(Γ−1) is the total fluid energydensity, and q_(α) is the species charge. The identity matrix isrepresented by II. In one example, adiabatic coefficient Γ=5/3 is used.Species temperature is determined by the relation p_(α)=n_(α)k_(B)T_(α),where k_(B) is the Boltzmann constant. The non-ideal terms, to bediscussed in below, are the stress tensors (Π_(α)), heat fluxes (h_(α)),and source terms S_(α) ^(n), S_(α) ^(m), and S_(α) ^(e), which representthe collisional sources of particles, momentum, and energy fromreactions and interactions between the species. In this example, thereare five equations in total—two scalar equations and one vectorequation—giving rise to the “five-moment” designation. All expressionsare in SI units unless otherwise noted.

The fluid equations are coupled to Maxwell's equations for magnetic (B)and electric (E) fields,

$\begin{matrix}{{{\nabla \times E} = {- \frac{\partial B}{dt}}},} & (4)\end{matrix}$ $\begin{matrix}{{{\nabla \times B} = {\mu_{0}\left( {j + {\epsilon_{0}\frac{\partial E}{\partial t}}} \right)}},} & (5)\end{matrix}$ $\begin{matrix}{{{\nabla \cdot B} = 0},} & (6)\end{matrix}$ $\begin{matrix}{{\nabla \cdot E} = {\frac{\rho_{c}}{\epsilon_{0}}.}} & (7)\end{matrix}$Here, μ₀ and E₀ are the permeability and permittivity of free space,respectively. The current density j=Σq_(α)n_(α)v_(α) and charge densityρ_(c)=Σq_(α)n_(α) (with the sums over a) provide coupling to the fluidequations. If the divergence constraints, Eqs. (6) and (7), aresatisfied in an initial value problem, they will remain satisfied,mathematically; equations (4) and (5) then completely describe theevolution of E and B. This strict mathematical guarantee is broken bythe presence of either numerical errors or domain boundaries, motivatingformulations that explicitly preserve the constraints. In the resultspresented herein, the divergence constraints are well-satisfied and nosuch special formulation is used.

To arrive at the 5M2F model, species are limited to ions and electrons,α=i, e. Collisional source terms arise only due to the Coulombscattering between ions and electrons. Specifically, the resultingsources are S_(α) ^(n)=0, S_(α) ^(m)=R_(α) ^(ie), and S_(α)^(e)=v_(α)·R_(α) ^(ie)+Q_(α) ^(ie), where R_(α) ^(ie) and Q_(α) ^(ie)are frictional exchange of momentum and energy, respectively. Physicaland numerical aspects of the 5M2F model are described at length inearlier work.

The model is closed using Chapman-Enskog-type closure, followingBraginskii. Stress tensors (Π_(α)) and heat fluxes (h_(α)) are specifiedaccording to the Braginskii formulation, allowing for arbitrarymagnetization, x_(α) =ω_(cα)τ_(α), where ω_(cα) is the cyclotronfrequency and T_(α) is the collision time for species α.

The momentum and thermal exchange terms (R_(α) ^(ie) and Q_(α) ^(ie))are dropped with justification as follows. The frictional momentumexchange modifies the bulk plasma momentum on a time scaleτ_(exch)≈(m_(i)/m_(e))τ_(e), where τ_(e) is the electron collisionalrelaxation time. Exchange of thermal energy occurs on the same timescale. If τ_(exch)/τ_(dyn)»1, where τ_(dyn) is the dynamical time scaleof interest, the frictional and thermal exchange terms may be omittedwithout loss of accuracy. The condition τ_(exch)/τ_(dyn)»1 is satisfiedin the FuZE-like plasma considered below, for example. Although theterms present no particular computational challenge, they are omitted toallow the presentation and analysis to focus on the viscosity and heatflux terms that are more important for m=0 instability behavior.

The heat fluxes areh _(α)=−κ_(⊥) ^(α)∇_(⊥) T _(α)±κ_(∧) ^(α) b×∇ _(⊥) T _(α),  (8)where the plus and minus signs are taken on the diamagnetic heat fluxterms (involving κ_(∧) ^(α)) for ions and electrons, respectively. Themagnetic field direction is b=B/B where B=|B|. Terms involving ∇_(∥)have been dropped because b is in the direction of symmetry. Theperpendicular thermal conductivities are

$\begin{matrix}{{\kappa_{\bot}^{\alpha} = {\frac{n_{\alpha}k_{B}^{2}T_{\alpha}\tau_{\alpha}}{m_{\alpha}}\frac{{\gamma_{1\alpha}^{\prime}x_{\alpha}^{2}} + \gamma_{0\alpha}^{\prime}}{\Delta_{\alpha}}}},} & (9)\end{matrix}$and the diamagnetic heat flux coefficients are

$\begin{matrix}{{\kappa_{\land}^{\alpha} = {\frac{n_{\alpha}k_{B}^{2}T_{\alpha}\tau_{\alpha}}{m_{\alpha}}\frac{x_{\alpha}\left( {{\gamma_{1\alpha}^{''}x_{\alpha}^{2}} + \gamma_{0\alpha}^{''}} \right)}{\Delta_{\alpha}}}},} & (10)\end{matrix}$where Δ_(α)=x_(α) ⁴+δ_(1α)x_(α) ²+δ_(0α). For electrons, the constantsare(γ_(0e)′,γ_(1e)′,γ_(0e)″,γ_(1e)″,δ_(0e),δ_(1e))=(11.92,4.664,21.67,2.5,3.7703,14.79),and for ions,(γ_(0i)′,γ_(1i)′,γ_(0i)″,γ_(1i)″,δ_(0i),δ_(1i))=(2.645,2.0,4.65,2.5,0.677,2.7).

The magnetization is calculated as x_(α)=ω_(cα)T_(α), where thecyclotron frequency is ω_(cα)=eB/m_(α). Assuming hydrogen ions, thespecies collision frequencies are

$\begin{matrix}{\tau_{e} = {\frac{3.5 \times 10^{11}}{\ln\Lambda}\frac{T_{e,{eV}}^{3/2}}{n_{i}}}} & (11)\end{matrix}$and

$\begin{matrix}{\tau_{i} = {\frac{2.12 \times 10^{13}}{\ln\Lambda}{\frac{T_{i,{eV}}^{3/2}}{n_{i}}.}}} & (12)\end{matrix}$

In these expressions, ln ∧ is the Coulomb logarithm, which is assumed toequal 10 herein, and the temperatures are in eV. Note that a part ofheat flux related to ion-electron friction is omitted here; in thisaxisymmetric formulation, that part would be h_(e)^(u)=3n_(e)kT_(e)u_(⊥)/(2ω_(ce)τ_(e)), where u=v_(e)−v_(i). This term isdropped under the assumption of large ω_(ce)τ_(e) over the bulk of the Zpinch.

The stress tensor is constructed from the rate-of-strain tensor,

=∇v+(∇v)^(T)−⅔

∇·v,  (13)and five viscosity coefficients, η₀, η₂, η₃, and η₄. The even viscositycoefficients are

$\begin{matrix}{{\eta_{0} = {0.96n_{i}k_{B}T_{i}\tau_{i}}},} & (14)\end{matrix}$ $\begin{matrix}{{\eta_{2} = {n_{i}k_{B}T_{i}{{\tau_{i}\left( {{\frac{6}{5}x_{i}^{2}} + 2.23} \right)}/\Delta_{\eta}}}},} & (15)\end{matrix}$ $\begin{matrix}{{\eta_{4} = {n_{i}k_{B}T_{i}\tau_{i}{{x_{i}\left( {x_{i}^{2} + 2.38} \right)}/\Delta_{\eta}}}},} & (16)\end{matrix}$where Δ_(η)=x_(i) ⁴+4.03x_(i) ²+2.33. The odd coefficients η₁ and η₃ arefound from η₂ and η₄ by replacing ω_(ci) with 2ω_(ci); that is,η₁=η₂(2x_(i)) and η₃=η₄(2x_(i)). Herein, a cylindrical coordinate systemis used with radial, azimuthal, and axial coordinates r, θ, and z,respectively. The magnetic field is taken to be strictly in theazimuthal direction. Under the assumptions of zero azimuthal velocity,and no variation in the azimuthal direction, the components of thestress tensor are

∏ θ ⁢ θ = - η 0 θ ⁢ θ , ( 17 ⁢ a ) ∏ z ⁢ z = - η 0 ⁢ 1 2 ⁢ ( z ⁢ z + r ⁢ r )  -η 1 ⁢ 1 2 ⁢ ( z ⁢ z - r ⁢ r ) - η 3 z ⁢ r , ( 18 ⁢ b ) ∏ r ⁢ r = - η 0 ⁢ 1 2 ⁢ (z ⁢ z + r ⁢ r )

 - η 1 ⁢ 1 2 ⁢ ( r ⁢ r - z ⁢ z ) + η 3 z ⁢ r , ( 19 ⁢ c ) ∏ z ⁢ r = ∏ r ⁢ z = -η 1 z ⁢ r + η 3 ⁢ 1 2 ⁢ ( z ⁢ z - r ⁢ r ) , ( 20 ⁢ d ) $\begin{matrix}{{\prod\limits_{z\theta}{= {\prod\limits_{\theta z}{= 0}}}},} & \left( {21e} \right)\end{matrix}$ $\begin{matrix}{\prod\limits_{r\theta}{= {\prod\limits_{\theta r}{= 0.}}}} & \left( {22f} \right)\end{matrix}$

As discussed by Braginskii, terms proportional to η₀ correspond tostress associated with compression or expansion of the plasma. Termsproportional to η₁ are associated with diffusion across the magneticfield with step size equal to the Larmor radius, and step frequency setby collisions. The coefficient η₃ is associated with gyroviscosity,which is the diamagnetic flux of momentum. Terms proportional to η₂ andη₄ have dropped out. Electron viscosity is omitted in thisimplementation (except for an isotropic viscosity applied for numericalpurposes) on the basis that for similar ion and electron temperatures,the electron viscosity coefficients η₀ ^(e), η₃ ^(e), and η₁ ^(e) aresmaller than their ion counterparts by approximately(m_(e)/m_(i))^(1/2), m_(e)/m_(i), and (m_(e)/m_(i))^(3/2), respectively,and under the further assumption that electron and ion velocitygradients are comparable.

Three types of corrections to the Braginskii transport coefficients areconsidered herein. The first is related to the assumption made inderiving the coefficients that time scales are long compared to particlecollision times. Stress in the r-z plane due to plasma compression orexpansion is regulated by the unmagnetized viscosity, η₀=0.96p_(i)τ_(i).As described by Braginskii, ∇·v<0 (compression) increases stress, while∇·v>0 (expansion) reduces stress. The magnitude of this stress isp_(i)τ_(i)|∇·v|. The physical mechanism is as follows. The continuityequation shows that ∇·v=−{dot over (n)}/n (ignoring gradients of n); interms of the dynamical time scale, {dot over (n)}/n=(δn/n)/τ_(dyn).Then, assuming flux frozen into the fluid, {dot over (n)}/n={dot over(B)}/B=(δB/B)/τ_(dyn). Assuming that the ion Larmor orbit size is smallcompared to the size of the region of compression or expansion,increased magnetic field gives increased transverse velocity andassociated transverse energy and stress. This effect manifests astemperature anisotropy, as observed in continuum kinetic simulations.The increased energy is partitioned between transverse and paralleldirections over a time set by τ_(i); this process is known asgyrorelaxation. With τ_(i)≈τ_(dyn), the transverse stress isapproximately pδB/B. For a δB/B=1, for example, the magnitude of thestress is similar to the isotropic pressure. When τ_(i)<T_(dyn), thestress is reduced due to fast equipartition. When τ_(i)>τ_(dyn),however, the as-derived effect is unphysically strong, giving a stresslarger than the isotropic pressure for δB/B=1. In the model implementedhere, a correction factor,

$\begin{matrix}{{f_{corr}^{\tau} = \left( {1 + \frac{{\hat{\tau}}_{i}}{\tau_{dyn}}} \right)^{- 1}},} & (23)\end{matrix}$is applied to the η₀ coefficient. Here, {circumflex over (τ)}_(i) is arepresentative ion collision time. The form used for this correction issimilar to the parallel heat flux correction commonly employed inBraginskii-based modeling in tokamaks. By using a representativecollision time, this correction is a global constant rather than onethat varies depending on local plasma parameters. This approach requiresa priori specification of τ_(dyn). For modeling m=0 Z-pinch instability,τ_(dyn) is set to the characteristic Alfvén time, τ_(A), defined ascharacteristic pinch radius divided by Alfvén speed (see completedefinition in herein).

The second and third corrections are related to the breakdown of theBraginskii model when Larmor radii are large compared to the lengthscale of interest. (There are not analogs of these corrections in thetokamak modeling community, in which small Larmor radius is typicallyassumed.) One is a global correction,

$\begin{matrix}{{{f_{corr}^{L} = {{{\frac{1}{2}\left\lbrack {1 - {\cos\left( \frac{\pi\ell}{2{\hat{r}}_{L\alpha}} \right)}} \right\rbrack}{where}{\hat{r}}_{L\alpha}} > \frac{\ell}{2}}},{and}}{{f_{corr}^{L} = {1{elsewhere}}},}} & (24)\end{matrix}$based on a representative Larmor radius, {circumflex over (τ)}_(Lα), andlength scale of interest,

. A correction with linear dependence on r_(L) would allow strongtransport even where the Larmor radius is larger than the feature size.The nonlinear dependence of Eq. (24) on Larmor radius is more physicallyplausible. For {circumflex over (r)}_(Lα)=

, the correction is ½, and for {circumflex over (r)}_(Lα)>

, the correction becomes stronger (e.g., f_(corr) ^(L)≈0.15 for{circumflex over (r)}_(Lα)=2

). This correction is applicable to the gyroviscosity (η₃) anddiamagnetic heat fluxes (κ_(∧) ^(α)), and also to the transverse (in ther-z plane) stress associated with η₀, which is linked to changes inLarmor orbit size due to gradients with scale length

. As for τ_(dyn),

is specified a priori. For modeling linear growth of a mode with knownwavenumber k, a sensible choice is

=k⁻¹. For nonlinear modeling, a sensible choice is

=k_(max) ⁻¹/2, where k_(max) is the wavenumber at which linear growth ismaximum. Because linear growth rates tend to fall for k>k_(max), thischoice ensures that the transport corrections are applied to thefast-growing modes.

The other correction for large Larmor radius pertains to the specialsituation near r=0, where magnetic field approaches zero, and Larmororbits are no longer simple helices. Moving radially from r=0, theLarmor orbit becomes finite, and may eventually reach a value thatmatches the radius. Using a critical radius, r_(crit), to approximatethis location, an r-dependent correction,

$\begin{matrix}{{{f_{corr}^{r} = {{{\frac{1}{2}\left\lbrack {1 - {\cos\left( \frac{\pi r}{r_{crit}} \right)}} \right\rbrack}{where}r} < r_{crit}}},{and}}{{f_{corr}^{r} = {1{elsewhere}}},}} & (25)\end{matrix}$is applied to reduce transport near r=0 where ion orbits are no longersimple helices. This correction is applied only to η₃ and κ_(∧) ^(i,e).As mentioned in the discussion above of Eq. (24), the nonlineardependence of f_(corr) ^(r), gives a strong cutoff where local Larmorradius exceeds the distance from the cylindrical axis. It is worthmentioning that the region of small magnetic field near r=0 is alsoexpected to affect transverse stress associated with η₀. Instead ofapproaching zero at r=0, however, η₀ should match the perpendiculartransport, η₁, at some level that represents the random walk processwithin the region of non-helical orbits. An associated correction is notattempted here.

The combination of these three correction factors is sufficient toexplore the basic effects of Braginskii transport on Z-pinch stability.As described further herein, examples of corrected coefficients arepresented in the context of a FuZE-like equilibrium.

The WARPXM modeling framework is used to solve the 5M2F model on anunstructured grid of triangles using a Runge-Kutta discontinuousGalerkin (RKDG) technique. WARPXM uses MPI parallelization, and thescalability of RKDG is suitable for problems with large dimensionality.

Because of the explicit time stepping employed, the fastest time scalemust be resolved. For hyperbolic phenomena, the time step size(Δt_(hyp)) is limited according to uΔt_(hyp)/h_(eff)≤C, where u is thewave speed, h_(eff) is the effective grid resolution, and C is theCourant number, which depends on the particular Runge-Kutta schemechosen, but is typically ≤1. In 5M2F applications, the speed of light,c=1/(μ₀ϵ₀)^(1/2), is often the fastest speed. To allow larger Δt_(hyp),c may be reduced by artificially increasing ϵ₀. To ensure systemstability, c must exceed the electron thermal speed,v_(Te)=(2k_(B)T_(e)/m_(e))^(1/2). Further increase of Δt_(hyp) ispossible by artificially enhancing electron mass to reduce v_(Te).Changes to c and m_(e) should be moderated to preserve physicalaccuracy. The allowable overall Δt is limited not only by hyperbolicphysics, but also by oscillatory and diffusive behavior. Oscillationswith angular frequency co require Δt_(0sc)≤0.5ω⁻¹. For diffusivebehavior, the time step must satisfy DΔt_(diff)/h_(eff) ²<C_(D), where Dis the relevant diffusivity and C_(D) is a diffusive Courant number oforder unity. This limit on Δt_(diff) can be severe, and it is sometimesuseful to artificially reduce diffusivities when physically justified.Accounting for all of these restrictions, at each time step,Δt=min(Δt_(hyp),Δt_(0sc),Δt_(diff)) is used.

Three additional details of WARPXM warrant explanation. First, specialattention is needed to accurately calculate the gradients needed in thefluxes associated with Braginskii closure, since the discretization ofprimary variables is discontinuous. To address this issue, theBassi-Rebay approach is used. Second, to accommodate the cylindricalcoordinate system, vector calculus operations are written in terms ofrectilinear derivatives and source terms. For example, the divergence ofthe vector A is often written as

${\nabla \cdot A} = {{\frac{1}{r}{\frac{\partial}{\partial r}\left( {rA_{r}} \right)}} + {\frac{1}{r}\frac{\partial A_{\theta}}{\partial\theta}} + {\frac{\partial A_{z}}{\partial z}.}}$

The azimuthal derivative is zero in certain embodiments of theaxisymmetric system described herein. The term involving the radialderivative can be rewritten to give

${\nabla \cdot A} = {\frac{\partial A_{r}}{\partial r} + \frac{\partial A_{z}}{\partial z} + {\frac{A_{r}}{r}.}}$

The infrastructure of WARPXM can then naturally handle the first twoterms, and the last term is included as a source term. Curl,divergence-of-tensor, and gradient-of-vector operations are also writtenin terms of rectilinear derivatives plus cylindrical source terms. Thethird detail is related to the spatial integration of these cylindricalsource terms. The basic DG method implemented in WARPXM integratessource terms using quadrature based on values at theLegendre-Gauss-Lobatto (LGL) nodes. Triangles have LGL nodes on theedges, and so some nodes are at r=0. Performing the quadrature thenrequires computing the source terms at r=0. Some of the cylindricalsource terms involve first derivatives of the primary variables dividedby r. Applying L'Hôpital's rule would require knowledge of the secondderivative. To avoid the need for second derivatives, the LGL quadratureis replaced with symmetric Gaussian quadrature which does not havequadrature points on the triangle edges, and thus avoids division by 0at r=0.

The diffuse Bennett Z-pinch equilibrium is the focus of the modelingpresented below. This equilibrium may be parameterized by a pinch radius(α) and plasma current (I_(p)). The azimuthal magnetic field (B_(θ)),axial current density (j_(z)), and total plasma pressure (p) are thengiven by

$\begin{matrix}{{B_{\theta} = {\frac{\mu_{0}I_{p}}{2\pi}\frac{r}{r^{2} + a^{2}}}},} & (26)\end{matrix}$ $\begin{matrix}{{j_{z} = {\frac{I_{p}}{\pi}\frac{a^{2}}{\left( {r^{2} + a^{2}} \right)^{2}}}},} & (27)\end{matrix}$ $\begin{matrix}{p = {\frac{\mu_{0}I_{p}^{2}}{8\pi^{2}}{\frac{a^{2}}{\left( {r^{2} + a^{2}} \right)^{2}}.}}} & (28)\end{matrix}$

FIG. 11 illustrates normalized radial profiles of magnetic field(B_(θ)), density (n), and temperature (T) in a Bennett equilibrium.

The total pressure consists of equal contributions from ion and electronpressures. A uniform temperature (T) is assumed such that density(n_(i)=n_(e)=n for shear-free equilibria) is proportional to pressure(p/2=nk_(B)T). Exactly half of the plasma mass and current are containedwithin r=a. Total pressure at r=a is equal to the magnetic pressure,B_(θ) ²/(2μ₀), at r=a. The magnetic field, density, and temperatureprofiles for this equilibrium, normalized by their respective peakvalues, are shown in FIG. 11 .

Momentum equilibrium in the 5M2F model requires

$\begin{matrix}{{\frac{\partial p_{i}}{\partial r} = {q_{i}{n_{i}\left( {E_{r} + {v_{iz}B_{\theta}}} \right)}}},} & (29)\end{matrix}$and

$\begin{matrix}{\frac{\partial p_{e}}{\partial r} = {q_{e}{{n_{e}\left( {E_{r} + {v_{ez}B_{\theta}}} \right)}.}}} & (30)\end{matrix}$

Except in cases with sheared flow (see below), equal and opposite ionand electron axial velocities are assumed such that the equilibriumE_(r) is zero. Thus, v_(iz)=−v_(ez)=j_(z)/(2en), where embodimentsherein assume singly charged ions with charge q_(i)=−q_(e)=e, where e isthe elementary charge. Because j_(z) and n have the same radialdependence, these velocities are radially uniform. The total currentsatisfies j×B=∇p, and the Lorentz force balances the radial pressuregradient for each species. To establish equilibrium with v_(iz)=0, astypically assumed in static MEM equilibrium, finite E_(r) would benecessary, along with attending charge separation to satisfy Gauss'sLaw. Some earlier work using the 5M2F model uses equilibria withv_(iz)=0 and E_(r)=0; in such cases, oscillations about the equilibriumstate are present, though they are presumably benign from the point ofview of growth rate determination.

Sheared flow is added to the equilibrium as follows. Ion axial velocityis v_(iz)=j_(z)/(2 en)+v_(sf). Sheared flow velocities that are linearor parabolic in radius are considered herein, i.e., v_(sf)(r)=v_(sf)^(a)r/a or v_(sf)(r)=v_(sf) ^(a)(r/a)², where v_(sf) ^(a) is the shearedflow velocity at r=a. The required electric field is determined by Eq.(29) with n_(f)=n, unchanged from the shear-free equilibrium. Gauss'sLaw is used to find n_(e). Axial current is unchanged from theshear-free equilibrium, and v_(ez) is set accordingly. Electron pressureis then determined by Eq. (30). For v_(sf) of practical interest (v_(sf)^(a)≤v_(A)), electron density and pressure differ only slightly («1%)from the shear-free values.

The perturbation used here is radially localized using an a knownapproach, and includes an optional phase shift. By including a phaseshift, perturbed mode shapes can closely match the final eigenmodestructure, which depends on the applied sheared-flow profile and otherplasma parameters. Perturbed density and current density are n+δn andj_(z)+δj_(z), withδn(r)=ϵn(r)cos(kz−ϕ ₀ r ^(ζ))e ^(−(r−a)) ² ^(/(2b) ² ⁾,  (31),andδj _(z)(r)=−ϵ/2j _(z)(r)cos(kz−ϕ ₀ r ^(ζ))e ^(−(r−a)) ² ^(/(2b) ²⁾,  (32)where k is the perturbation wavenumber. The phase shift is determined bythe parameter ϕ₀ and the factor r^(ζ); ζ=1 and 2 are used in simulationswith linear and parabolic sheared flow, respectively. The radiallocalization uses the parameter b=a/3. The perturbations added to ionand electron velocity are δv_(iz)=−δv_(ez)=δj_(z)/(2en). The equilibriummagnetic field is unchanged, so Faraday's Law is out of balance suchthat electric field immediately begins evolving in response to theperturbation.

The computational domain is rectangular in the r-z plane. The axiallength matches the perturbed wavelength, i.e., L_(z)=2π/k. The radialextent of the domain is 4a. This domain setup matches previous work onm=0 instability analysis. The axial boundaries are periodic. At r=0,standard axisymmetry boundary conditions are used: the radial andazimuthal components of vector quantities are zero, while scalars andaxial components of vector quantities have no radial variation. At r=4a,perfect-slip conducting wall boundary conditions are applied. That is,radial velocity, radial magnetic field, and axial electric field arezero. Density, pressure, axial velocity, radial electric field, andaxial magnetic field have no radial variation.

Several characteristic quantities are used in the analysis anddiscussion below. The characteristic Alfvén velocity isv_(A)=B_(θ,pk)/(n_(pk)m_(i)μ₀)^(1/2), where B_(θ,pk) and n_(pk) are thepeak magnetic field and number densities. Characteristic time is definedas τ_(A)=a/v_(A). Ion thermal speed is v_(Ti)=(2k_(B)T/m_(i))^(1/2). IonLarmor radius is r_(Li)=m_(i)v_(Ti)/(eB_(θ,pk)). The thermal speed isrelated to the Alfvén velocity as v_(Ti)=√{square root over (2)}v_(A).Hydrogen ions are assumed.

For a given choice of a and I_(p), the pressure is determined, but thedensity and temperature are not yet specified. Density and temperatureare established by specifying the ratio of pinch size to ion Larmorradius, a/r_(Li). Using the definition of v_(Ti), solving for T_(i)gives

$\begin{matrix}{T = {T_{e} = {T_{i} = {\frac{\left( {aeB_{\theta,{pk}}} \right)^{2}}{2m_{i}{k_{B}\left( {a/r_{Li}} \right)}^{2}}.}}}} & (33)\end{matrix}$Ion and electron density follow from the relation nk_(B)T=p/2.

In the ideal 5M2F model, normalized dynamics, such as the normalizedinstability growth rate γτ_(A), depend on the choice of a/r_(Li), butspecific choices of a and I_(p) are not important. Braginskii transport,however, depends on the plasma properties (density, temperature, andmagnetic field), and in the applications discussed below of 5M2F withBraginskii closure, a, I_(p), and a/r_(Li) are all specified.

Sotnikov et al. [V. I. Sotnikov, I. Paraschiv, V. Makhin, B. S. Bauer,J. N. Leboeuf, and J. M. Dawson, “Linear analysis of sheared flowstabilization of global magnetohydrodynamic instabilities based on theHall fluid model,” Phys. Plasmas 9, 913 (2002)] apply a linearizedHall-MHD model to study growth of the m=0 instability in Bennettequilibria, with various Hall parameter strength. The Hall effect isparameterized using ϵ_(Sot.)=c/(ω_(pi)R), whereω_(pi)=[n_(i0)e²/(ϵ₀m_(i))]^(1/2) is the ion plasma frequency, and R isthe radius of the modeled domain, and n_(i0) is the ion density at r=0.To relate ϵ_(Sot.) to the parameter a/r_(Li), note that Sotnikov et al.use R=3a. Using r_(Li) and v_(Ti) introduced earlier, the relationshipis found to be a/r_(Li)=(3√{square root over (2)}ϵ_(Sot.))⁻¹. TheSotnikov et al. cases with ϵ_(Sot.)=0.1 and 0.01 are used as benchmarkcases; these correspond to a/r_(Li)=(3√{square root over(2)}ϵ_(Sot.))⁻¹≈2.357 and 23.57. Noting that the characteristic timeused by Sotnikov et al., τ_(Sot.)=R/v_(Ti), is related to Alfvén time byτ_(A)=τ_(Sot.)√{square root over (2)}/3, the growth rate at ka=10/3 withno sheared flow for ϵ_(Sot.)=0.1 is γτ_(A)=1.27, and with ϵ_(Sot.)=0.01is γτ_(A)=0.80. With ϵ_(Sot.)=0, in which case the Hall-MHD modelreduces to ideal MHD, γτ_(A)=0.73.

In ideal 5M2F simulations without sheared flow, close agreement with MHDand Hall-MHD results is seen. Linear growth rates for the m=0instability are determined from the linear growth phase in the nonlinearsimulations. With normalized axial wavenumber ka=10/3, growth rates fora/r_(Li) ranging from 4 to 200 are shown in FIG. 12A. For thesesimulations, the mass ratio is m_(e)/m_(i)=1/100 and the perturbationlevel is ϵ=10⁻³. Growth rates are found by considering the change involume-integrated radial kinetic energy of the ion fluid(KE_(i,rad.)=∫_(V)dVm_(i)n_(i)v_(ir) ²/2), where v_(ir) is the ionradial velocity, during a period of exponential growth from time t₀ tot₁. The growth rate is computed as

$\begin{matrix}{\gamma = {\frac{\ln\left\lbrack \frac{K{E_{i,{{rad}.}}\left( t_{1} \right)}}{K{E_{i,{{rad}.}}\left( t_{0} \right)}} \right\rbrack}{2\left( {t_{1} - t_{0}} \right)}.}} & (34)\end{matrix}$

An example time trace of ln(KE_(i,rad)/ME₀), where ME₀ is the initialmagnetic energy, is shown in FIG. 12B.

FIGS. 12A and 12B illustrate ideal 5M2F results at ka=10/3. FIG. 12Aillustrates growth rates for a/r_(Li) ranging from 4 to 200. At largea/r_(Li), the 5M2F growth rate agrees with result found by Sotnikov etal. for pure MHD (dashed line). For smaller a/r_(Li), the 5M2F resultsare close to the corresponding Sotnikov et al. results. FIG. 12Billustrates an example time trace of integrated radial ion kineticenergy, normalized by initial magnetic energy. Vertical dotted linesbracket the period over which linear growth is measured, and a dashedline shows the measured exponential growth.

Results for a relatively large Larmor radius regime are shown in FIG.13A, again for ka=10/3. For a/r_(Li)≤2, the result usingm_(e)/m_(i)=1/100 is similar to the result with real mass ratiom_(e)/m_(i)=1/1836, deviating by up to ≈30% at a/r_(Li)≈1.2. Fast growthappears in the region with a/r_(Li)≤1.3. As explored further below, thisfast growth is attributed to an electron drift instability. Radial ionkinetic energy growth is plotted in FIG. 13B for a/r_(Li)=1.2. Rapidkinetic energy growth is seen at t/τ_(A)≈2, near the end of thesimulation. Inspection of the solution shows that the rapid growth isdue to the development of a high-k mode with wavelength set by thecomputational grid. Refining the grid results in a faster, higher-kmode, and is thus counterproductive to the goal of identifying thegrowth rate of the perturbed ka=10/3 mode. The linear growth period ofthe perturbed mode is very short for simulations with a/r_(Li)≤1, makingit difficult to precisely determine the simulated growth rate; growthrates found for cases with a/r_(Li)≤1.3 have errors on the order of 10%.

FIGS. 13A and 13B illustrate ideal 5M2F results at ka=10/3, focusing onsmall a/r_(Li). FIG. 13A illustrates growth rates for a/r_(Li) rangingfrom 0.75 to 4 for two electron-ion mass ratios. For a/r_(Li) between1.5 and 4, growth rates vary slowly. For a/r_(Li) less thanapproximately 1.1, faster growth is observed. FIG. 13B illustrates anexample time trace of integrated radial ion kinetic energy, normalizedby initial magnetic energy.

Additional scans of a/r_(Li) are performed at ka=20/3, 40/3, and 80/3.All of these simulations use m_(e)/m_(i)=1/1836. Results are shown inFIG. 14 . The region with fast growth due to the electron driftinstability persists in the region a/r_(Li)≤1.8. A region with no growthemerges where a/r_(Li)≥1.8, and expands as the perturbed wavenumber isincreased. For a given ka, growth goes to zero as r_(Li) approaches themode wavelength. In terms of ka, a value of a/r_(Li) below which dampedgrowth is expected may be approximated as (a/r_(Li))_(damp)≈ka/(2π). Atka=40/3, for example, (a/r_(Li))_(damp)≈2.11, consistent with the resultseen in FIG. 14 . The zero-growth region extends from (a/r_(Li))_(damp)to a/r_(Li)≈1.8, where an electron drift instability appears.

FIG. 14 illustrates ideal 5M2F results with several perturbedwavenumbers. A fast-growing mode persists at a/r_(Li)≤1.8, but a regionwith no growth appears and widens as the perturbed wavenumber increases.

For m_(e)/m_(i)=1/100 and ka=10/3, density structures of the linearlygrowing modes are shown in FIG. 15 for a/r_(Li)=4, 10, and 50. At larger_(Li), a slight drift of the mode structure in the +z direction isvisible; at a/r_(Li)=4, the total shift is about ⅛ of the axialwavelength. A tilt is seen at large r_(Li), with the structure atgreater radii shifted in the +z direction more than the structure atsmaller radii. Spatial resolution for the results in FIG. 15 is 64×16(radial×axial) cells, with third-order spatial accuracy.

FIG. 15 illustrates ideal 5M2F mode structures at a/r_(Li)=4, 10, and50. The base-10 logarithm of the change in density, normalized to thepeak equilibrium density is plotted in the r-z plane. At a/r_(Li)=50(MHD-like), a single lobe, indicated by a dashed oval, dominates theradial structure. At smaller a/r_(Li), two lobes are present, indicatedby dashed ovals in the case with a/r_(Li)=4: a smaller lobe is presentnear the axis, and is not aligned with the main lobe.

The growth rate values shown in FIG. 12A have errors ≤1%. Three knownsources of error are considered to arrive at this conclusion:

Spatial resolution: The scan uses a base spatial resolution 40×10(radial×axial) cells, with third-order spatial accuracy. At a/r_(Li)=4,10, and 50, the convergence behavior with respect to spatial resolutionis studied. In each case, error at the base resolution is <0.1%. Thefeature size, as seen in FIG. 15 is half of the axial domain and isrepresented by five cells. With a third order representation (which usesquadratic polynomials), the accuracy is very good. The radialrepresentation of the modes is as good or better than the axialrepresentation.

Mass ratio: A enhanced electron mass m_(e)/m_(i)=1/100 is used in thescan. Simulations that use real mass ratio (m_(e)/m_(i)=1/1836) are runat a/r_(Li)=4, 10, and 50, and used to determine the error associatedwith the enhanced mass. Errors are 1.0%, 0.6%, and 0.8% at a/r_(Li)=4,10, and 50, respectively. With m_(e)/m_(i)=1/100, the combinedion-electron fluid density in the 5M2F model is 1% larger than that ofthe MHD fluid, so errors on the order of 1% are expected. In situationswith strong two-fluid effects, enhanced electron mass can cause largererror, as seen in FIG. 13A.

Speed of light: The speed of light is set to 3v_(Te) in the scan. Ata/r_(Li)=4, 10, and 50, simulations are run with the speed of lightdoubled, and redoubled, up to 12v_(Te) in each case. Over this range,variations of the measured growth rate are <0.5% at all three values ofa/r_(Li), indicating that the reduced electromagnetic wave speed hasonly a minor effect on the modeled instabilities.

Ideal 5M2F simulations of the ka=10/3 mode with linear and parabolicsheared-flow profiles give results similar to prior work using MHD andHall MHD. FIGS. 16A and 16B show the m=0 growth rates in scans of thesheared-flow strength, v_(sf). Growth rates from the 5M2F model witha/r_(Li)=50, are compared with MHD results, and 5M2F with a/r_(Li)=2.357are compared with Hall-MHD results that use an equivalent Hallparameter. As shown, MHD-like 5M2F modeling at a/r_(Li)=50 givesqualitatively similar results to ideal MHD modeling with linear andparabolic sheared-flow profiles. Complete stabilization occurs withpinch-edge velocity is v_(sf) ^(a)≈0.4v_(A). In plasmas witha/r_(Li)=2.357, 5M2F simulations with linear sheared flow indicates thatstabilization requires v_(sf) ^(a)/v_(A)>0.6. (Note that comparablesimulations of linear sheared flow with Hall MHD are not readilyavailable.) With parabolic sheared flow, 5M2F modeling suggestsstabilization of plasmas with a/r_(Li)=2.357 at v_(sf) ^(a)/v_(A)>0.5,similar to the linear sheared-flow case. The 5M2F modeling agrees wellwith the MHD and Hall-MHD results.

FIGS. 16A and 16B illustrate ideal 5M2F results with stabilization bylinear and parabolic sheared flows, respectively. All simulations haveka=10/3. Growth rates found with MHD and Hall-MHD modeling are shown forcomparison. The scans shown in FIG. 16A include one scan with reversedsheared flow, labeled “neg. v_(sf) ^(a)”; all other scans in FIG. 16Ahave positive v_(sf) ^(a).

The prior work with Hall MHD (and also the PIC results discussed herein)used positive sheared flow, i.e., positive values of v_(sf) ^(a).Anticipating that two-fluid effects will depend on sheared-flowdirection, a scan is done with negative v_(sf) ^(a), i.e., reversedsheared flow, as shown in FIG. 16A. The stabilizing effect of shearedflow is reduced when flow is reversed, and of v_(sf) ^(a)>v_(A) isrequired for stabilization. As a/r_(Li) becomes large, the growth ratesseen with sheared flow—positive or negative—will approach the MHDresults, which are independent of sheared-flow direction.

FIG. 17 illustrates mode structures in ideal 5M2F modeling with shearedflow. Results are shown for linear (top row) and parabolic (bottom row)sheared flow, for MHD-like (a/r_(Li)=50) and large-Larmor-radius(a/r_(Li)=2.357) plasmas as indicated. All cases use positive shearedflow (in the sense discussed in the text) except for one case withnegative linear sheared flow labeled as “neg.” In all cases, thesheared-flow speed at the pinch edge is |v_(sf) ^(t)|=0.3v_(A).

Sheared flow stretches the exponentially growing mode axially. FIG. 17shows mode structures for simulations with sheared-flow speed |v_(sf)^(a)|=0.3v_(A). Comparing linear and parabolic results for a givena/r_(Li), parabolic sheared flow restricts the mode structure to smallerradius. For linear sheared flow with a/r_(Li)=2.357, results are shownfor both positive and negative sheared flow. In the case with negativesheared flow, alignment of the near-axis lobe of the structure with themain lobe seems to promote the observed larger growth rate.

To accurately capture steady exponential growth in the presence of suchstretching, the resolution and perturbation size and shape are chosencarefully. Full development of the mode structure takes approximately5τ_(A) or longer, depending on plasma parameters and sheared flowdetails. If the perturbation level is ϵ=10⁻³ as used in the shear-freesimulations, nonlinearities may affect mode growth before fulldevelopment. Therefore, in these sheared-flow simulations, a smallerperturbation is used. Higher resolution is also used in these cases,both to limit noise, which can obscure growth of the small perturbation,and to minimize unintended seeding of short wavelength modes that cangrow quickly and disturb development at the perturbed axial wavenumber.Short wavelength modes are less problematic in the cases witha/r_(Li)=2.357, since strong two-fluid effects suppress high-k growth.To further mitigate difficulties with high-k growth and nonlinearities,a phase shift, as discussed elsewhere herein, is included. Anappropriate value of the phase shift parameter, ϕ₀, is found byexecuting a preliminary simulation with ϕ₀=0, observing the modestretching in the late stages of the simulation prior to interruption ofgrowth by nonlinearities or high-k mode growth. For a second and finalsimulation, ϕ₀ is chosen such that the shape of the initial conditionroughly matches the stretched mode seen in the preliminary simulation.FIGS. 18A-18D illustrate an example of mode growth with and without aphase-shifted initial condition. Without phase shift, the modestretching results in a steadily decreasing growth rate before thesimulation is finally disrupted by high-k growth, observable at t18τ_(A). By including a phase shift, the simulation rapidly enters aperiod of linear growth. The lines in FIGS. 18A and 18C represent datacomputed at τ_(A)/10 intervals. In FIGS. 18B and 18D, the values ofγτ_(A) at times t are computed using Eq. (34) with t₁=t and t₀=t−2τ_(A).

The MHD-like 5M2F simulations (a/r_(Li)=50) all use 64×16 (radial×axial)cells with fifth-order elements and ϵ=1×10⁻⁵. Large-r_(Li) simulations(a/r_(Li)=2.357) use 64×16 (radial×axial) cells with fourth-orderelements and ϵ=1×10⁻⁷. The phase shifts are given in TABLE I.

TABLE I Settings used for ideal 5M2F simulations with sheared flow.Phase shift parameters ϕ₀ are given for each run with sheared flow. Forlinear sheared flow, the overall phase shift is ϕ₀r, while for parabolicsheared flow, it is ϕ₀r²; see Eqs. (31) and (32). a/r_(Li) sheared flowtype ν_(sf) ^(a)/ν_(A) ϕ₀ 50 none 0 0   . linear 0.1  π/2 . . 0.2  π . .0.3 2π  . . 0.35 3π  2.357 none 0 0   . linear 0.1  π/2 . . 0.2  3π/4 .. 0.3  π . . 0.4  3π/2 . . 0.5  7π/4 . . 0.6 2π  2.357 linear −0.1 0   .. −0.2  −π/4 . . −0.3  −π/2 . . −0.4 −3π/4 . . −0.6 −π  . . −0.8 −3π/250 parabolic 0.1  π/2 . . 0.2  3π/4 . . 0.3  3π/2 . . 0.35 2π  2.357parabolic 0.1  π/4 . . 0.2  π/2 . . 0.3  3π/4 . . 0.4  7π/8 . . 0.5  π

Deviation from the exact linear growth rate in these simulations isestimated to be no more than 10% and generally a few percent or less.This level of error is higher than for the shear-free results primarilydue to the challenges associated with simultaneously avoiding high-kmodes and nonlinearities.

FIGS. 18A-18D illustrate mode growth behavior in the 5M2F model with andwithout an initial phase shift (effectively a tilt) in the perturbation.Results are for simulations in an MHD-like regime (a/r_(Li)=50),including linear sheared flow with pinch edge velocity v_(sf)^(a)=0.3v_(A). FIG. 18A shows evolution of radial ion kinetic energynormalized by initial magnetic energy. FIG. 18B shows the normalizedgrowth rate, γτ_(A), derived from the kinetic energy growth at 0.1τ_(A)intervals. The approach used to find each plotted value (x) is discussedin the main text. The red line shows the 1−τ_(A) moving average of theindividual values. FIGS. 18C and 18D are similar to FIGS. 18A and 18B,but for a case with perturbation phase shift ϕ₀r with ϕ₀=2π.

In recent research using PIC modeling, the m=0 instability wasconsidered in a FuZE-like equilibrium. In the discussion below, theequilibrium is presented, and details related to implementing Braginskiitransport in this equilibrium are provided below. Also below, 5M2Fmodeling with Braginskii transport is compared with PIC results.

A Bennett equilibrium is used to represent a typical FuZE plasma. Asdiscussed in detail above, the normalized profiles in FIG. 11 may begiven dimensions by choosing I_(p), a, and a/r_(Li). Following previouswork, the FuZE-like equilibrium has a/r_(Li)=5.825, a=0.91 mm andI_(p)=300 kA. These choices give B_(θ,pk)=33.0 T, n_(pk)=4.25×10²⁴ m⁻³,and T=1.27 keV. The associated characteristic time and Alfvén velocityare τ_(A)=2.61 ns and v_(A)=3.49×10⁵ m/s.

The Braginskii closure model, including kinetic corrections, ispresented herein. Here, the transport coefficients, before and aftercorrection, are considered specifically in the FuZE-like equilibrium.FIG. 19 shows corrected and uncorrected viscous diffusivities.Corrections are made according to Eqs. (23), (24), and (25). In Eq.(23), {circumflex over (τ)}_(i) is τ_(i) computed at peak density, andτ_(dyn)=τ_(A). In Eq. (24), {circumflex over (r)}_(Li) is r_(Li), thecharacteristic ion Larmor radius discussed herein, and

is chosen to be 0.2a, corresponding to k⁻¹, with ka=5. In Eq. (25),r_(crit)=0.5a, which is comparable to the location r=0.31a at which theion Larmor radius matches the radius itself. Diffusivities are found bydividing the viscosity coefficients by mass density (ρ=m_(i)n_(i)).Braginskii thermal transport coefficients, discussed below, are alsoexpressed as diffusivities to facilitate comparison. The correctedcoefficient η₀ ^(*)/ρ is found as η₀ ^(*)/ρ=f_(corr) ^(τ)f_(corr)^(L)η₀/ρ. A diffusivity limit, D_(lim) is also enforced, with D_(lim)equal to η₀ ^(*)/ρ at r=a. The diffusivity limit is imposed for purelynumerical reasons: diffusion associated with η₀ ^(*)/ρ typically setsthe maximum time step that can be used in simulations. By imposingD_(lim), the dynamics of interest are preserved, but the time step canbe increased significantly. The corrected coefficient η₃ ^(*)/ρ is foundas η₃ ^(*)/ρ=f_(corr) ^(L)f_(corr) ^(e)η₀/ρ. The corrected coefficientη₁ ^(*)/ρ is found as η₁ ^(*)/ρ=max(η₀ ^(*)/ρ,η₁/ρ).

FIG. 19 illustrates momentum diffusivities in the FuZE-like equilibrium.Uncorrected diffusivities (coefficients without *) have features thatare unphysical and/or numerically challenging. Diffusivities arecorrected (coefficients with *) to address these issues as described inthe main text. Corrections are made per Eqs. (23), (24), and (25) with

=0.2a and r_(crit)=0.5a.

FIG. 20 shows corrected and uncorrected ion thermal diffusivities.Corrections are made using

=0.2a and r_(crit)=0.5a, as for momentum diffusivities. The diffusivitylimit used for momentum diffusivity is also imposed, though it affectsκ_(⊥) ^(i) in only a small region near r=0. Electron thermaldiffusivities are not shown; of the Braginskii electron transport terms,only the diamagnetic heat flux term is retained, and the associatedcoefficient κ_(∧) ^(e) is almost identical to its ion counterpart exceptvery near r=0. In that region, radial variation of κ_(∧) ^(e) is moreextreme than κ_(∧) ^(i). A correction factor similar to f_(corr) ^(L) inEq. (24) is not applied, since the characteristic electron Larmor radiusis small compared to

. The location at which the electron Larmor radius matches the radiusitself is 0.09a, so using r_(crit)=0.2a would be reasonable. Using thissmall r_(crit) and applying f_(corr) ^(r) per Eq. (25) yields κ_(∧)^(e*)/(n_(e)e) peaking near 300 m²·s⁻¹ at r=0.15. r_(crit)=0.5a ischosen to give a more numerically tractable profile of κ_(∧)^(e*)/(n_(e)e), with a peak near 120 m²·s⁻¹ at r=0.4. At r=0.4 andbeyond, κ_(∧) ^(e*)/(n_(e)e) is nearly identical to the uncorrectedκ_(∧) ^(i)(n_(i)e) shown in FIG. 20 .

FIG. 20 illustrates ion thermal diffusivities in the FuZE-likeequilibrium. Uncorrected diffusivities (coefficients without *) havefeatures that are unphysical and/or numerically challenging.Diffusivities are corrected (coefficients with *) to address theseissues as described in the main text. Corrections are made per Eqs. (24)and (25) with

=0.2a and r_(crit)=0.5a.

Simulations indicate that, of all of the Braginskii transport terms, thetransverse stress regulated by η₀ has the strongest influence on m=0instability growth. To study the effect of η₀ in comparison with thegyroviscosity and diamagnetic heat flux, three series of simulations aredone at fixed wavenumber ka=5. Results are presented in FIG. 21 . Forthe purpose of scanning the strength of η₀, the product of the twoglobal correction factors described above, i.e., f_(corr) ^(τ)f_(corr)^(L), is replaced with a single multiplier, f_(η0), which is used tocalculate a modified unmagnetized viscosity: η₀ ^(*)=f_(η0)η₀. In thefirst series, the full Braginskii model noted above is applied, with ther_(crit)-based corrections applied to gyroviscosity and diamagnetic heatflux coefficients, but no correction for large Larmor radius is used forthose coefficients. In the second series, gyroviscosity is omitted, butdiamagnetic heat flux retained. In the third series, diamagnetic heatflux is omitted, but gyroviscosity retained. Using the full Braginskiimodel, as f_(η0) approaches 0.1, the modeled growth rate approaches thePIC result. At small f_(η0), the gyroviscous and diamagnetic heat fluxcontributions each give a moderate (≈10%) reduction of the growth ratefrom the ideal 5M2F value. As f_(η0) approaches 0.1, the role ofgyroviscosity is reduced (i.e., the “no gyrovisc.” result approaches the“full Brag.” result in FIG. 21 ), but that of diamagnetic heat fluxremains moderate (i.e., the “no dia. heat flux” result remains 10%higher than the “full Brag.” result in FIG. 21 ). The 5M2F results shownin FIG. 21 use real electron mass (m_(i)/m_(e)=1836), perturbation sizeϵ=10⁻³ and 40×10 (radial×axial) cells with third order.

FIG. 21 illustrates 5M2F m=0 instability growth rates at ka=5 as afunction of an overall multiplier, f_(η0), on the unmagnetized viscositycoefficient, η₀. Results are shown for the full Braginskii model, thefull model minus gyroviscous term, and the full model minus thediamagnetic heat flux terms. The result at ka=5 from PIC modeling byTummel et al. [K Tummel, D. P. Higginson, A. J. Link, A. E. W. Schmidt,D. T Offermann, D. R. Welch, R. E. Clark, U. Shumlak, B. A. Nelson, R.P. Golingo, and H. S. McLean, “Kinetic simulations of sheared flowstabilization in high-temperature Z-pinch plasmas,” Phys. Plasmas 26,062506 (2019)] is included for comparison.

Examining the dependence of growth rates on k shows that applying the5M2F model with corrected Braginskii transport can give results thatshare some features of PIC results. As shown in FIG. 21 , the fullBraginskii model with f_(η0)=0.06 yields a growth rate similar to PIC.(Note that the overall factor f_(corr) ^(τ)f_(corr) ^(L)=0.018 is usedfor the corrected diffusivity η₀ ^(*)/ρ shown in FIG. 19 .) Scanning kawith fixed f_(η0)=0.06, γτ_(A) rises with ka and shows no sign ofreaching a peak even at ka=23, as shown in FIG. 22 . In this scan withf_(η0)=0.06, no correction for large Larmor radius is used. The reasonfor the rising γτ_(A) at large ka is unclear. Additional simulationsshow that the rise at high-k is persists even if gyroviscosity anddiamagnetic heat fluxes are omitted. The result demonstrates that usinga large η₀—unphysically large, as discussed above—is not a suitableapproach for reproducing PIC results across a wide range of ka. Resultsare also shown for a scan employing the complete size-based correctionmodel discussed above. In that scan, f_(η0)=f_(corr) ^(τ)f_(corr) ^(L),with

=K⁻¹, and f_(corr) ^(L) is applied to η₃ and κ_(∧) ^(i). With thesize-based model, decreasing growth at high-k is recovered. Notably,however, the result is similar to the ideal 5M2F result, which is shownfor comparison. Finally, a scan is done with transport correctionsdesigned for nonlinear modeling; see discussion of rationale herein. Inthe nonlinear setup,

=0.1a and a minimum diffusivity, D_(min)=0.01av_(A) is applied to η₁ andκ_(⊥) ^(i,e). Isotropic electron viscosity, based on D_(min), is alsoincluded. Corrections are included per Eqs. (24) and (25) withr_(crit)=0.5a and fixed feature size

=0.1a. With this nonlinear setup, growth rates are reduced, giving aresult more similar to the PIC result than when real electron mass andD_(min)=0 are used. The mass ratios used for the simulations in FIG. 22are shown in the legend. The perturbation is again ϵ=10⁻³, andresolution is 40×10 cells with third order.

FIG. 22 illustrates 5M2F m=0 instability growth rates at modewavenumbers ranging from ka=5 to 23. Results from PIC modeling by Tummelet al. are included for comparison. Using constant viscosity correctionfactor f_(η0)=0.06, growth rates increase unphysically at high k.Computing f_(η0) with the size-based model using

=k⁻¹, a peak growth rate is seen at ka≈12, and growth rates decreasewith increasing k as in the ideal case. Also shown is the result withthe corrected transport model used for nonlinear modeling: enhancedelectron mass (m_(i)/m_(e)=1/100), f_(η0) based on a fixed size

=0.1a, and a diffusivity minimum D_(min)=0.01av_(A). The nonlinear setupgives growth rates that are most similar to PIC results.

The model described above is more rigorously nonlinear, but resultspresented so far have considered only the linear regime of instabilitygrowth, i.e., growth resulting in small deviations from the equilibriumstate. In this section, simulations that track plasma evolution into thenonlinear regime are considered. To enable this nonlinear modeling, acombination of high resolution and artificially large perpendiculartransport is used. The numerical methods readily available via WARPXM donot inherently provide strong numerical dissipation where gradientsbecome steep, and the polynomial representation is susceptible tounphysical oscillations (similar to the Gibbs phenomenon) that canterminate simulations. Artificial perpendicular viscosities and thermalconductivities help to relax sharp gradients, and high resolution helpsto resolve them.

In the simulations presented here, a minimum diffusivityD_(min)=0.01av_(A)≈3.2 m²·s⁻¹ is applied to η₁ and κ_(⊥) ^(i,e), and isused for isotropic electron viscosity: η₀ ^(e)/ρ_(e)=η₁^(e)/ρ_(e)=D_(min)av_(A), with ρ_(e)=m_(e)n_(e). As seen in FIGS. 19 and20 , η₁/ρ and κ_(⊥) ^(i)(n_(i)e) are approximately 0.04 and 0.27 m²·s⁻¹,respectively, at r=a. These correspond to Reynolds and Péclet numbersRe_(⊥) ^(a)≡av_(A)/0.04≈7900 and Pe_(⊥) ^(a)≡av_(A)/0.27≈1200,respectively. The artificial minimum diffusivity reduces these to Re_(⊥)^(a*)=Pe_(⊥) ^(a*)=100. Kinetic corrections per Eqs. (24) and (25) aremade with characteristic time and feature size are set to τ_(dyn)=τ_(A)and

=0.1a. An additional correction, specific to this nonlinear modeling, ismade to prevent spurious numerical behavior near the outer radialboundary associated with the diamagnetic heat and momentum flux terms. Afactor f_(trunc) is applied to truncate η₃ and κ_(∧) ^(i,e) in the outerregion of the plasma. Specifically,

${f_{trunc} = {{\frac{1}{2}\left\{ {1 + {\cos\left\lbrack \frac{\pi\left( {r - r_{trunc}} \right)}{r_{max} - r_{trunc}} \right\rbrack}} \right\}{where}{}r} > r_{trunc}}},$andf _(trunc)=1 elsewhere,  (35)where r_(max)=4a and r_(trunc)=2a. The difficulty with the diamagneticflux is associated with boundary conditions on derivatives of the 5M2Fvariables. Well-behaved boundary conditions are possible, in principle;however, the truncation above enables accurate simulation of nonlinearplasma dynamics in the region of interest, r≤2a.

Using the same initial condition as used for the PIC comparison at ka=5,but with parabolic sheared flow applied, simulations are done withresolution increased to 96×24 (radial×axial) cells with fourth-orderelements. No phase shift is used in the perturbation. A range ofparabolic sheared flow strength is studied, with v_(sf) ^(a)/v_(A)=0,0.25, 0.5, and 0.75. For the cases with v_(sf) ^(a)/v_(A)=0 and 0.5,evolutions of the two-dimensional density profiles are respectivelydepicted in FIGS. 1 and 3 . As the sheared flow speed increases, radialejection of plasma is limited. This is illustrated further in FIG. 23 ,where normalized pinch ion inventory and total thermal energy areplotted as a function of time. Pinch ion inventory and thermal energyare defined as

$\begin{matrix}{{{N(t)} = {\int_{0}^{a}{dr{\int_{0}^{L_{z}}{dzn_{i}2\pi}}}}},} & (36)\end{matrix}$and

$\begin{matrix}{{W(t)} = {\int_{0}^{a}{dr{\int_{0}^{L_{z}}{dz\frac{1}{\gamma - 1}\left( {p_{i} + p_{e}} \right)2\pi{r.}}}}}} & (37)\end{matrix}$The normalized quantities are N(t)/N(t=0) and W(t)/W(t=0). In theshear-free case, the pinch has lost over 50% of the initial inventory byend of simulation, while in the case with v_(sf) ^(a)/v_(A)=0.5, 30% islost. Thermal energy loss is 20% in the shear-free case and 10% in thev_(sf) ^(a)/v_(A)=0.5 case.

FIG. 23 illustrates ion density profiles for nonlinear simulationsstarting from FuZE-like equilibria with seeded mode ka=5. The profilesare repeated three times axially and reflected across r=0. In the casewith zero sheared flow (top row), by t=10τ_(A), radial jet-likestructures have developed. With v_(sf) ^(a)/v_(A)=0.5 (bottom row), thegrowth is delayed, instability structures are sheared and limited inradial extent by the shear, and a relaxed near-equilibrium state isestablished by t=35τ_(A).

FIG. 24 illustrates normalized ion inventory and thermal energy withinr=a, normalized by initial inventory and energy, for FuZE-like nonlinearsimulations with v_(sf) ^(a)/v_(A)=0, 0.25, 0.5, and 0.75. In all cases,ka=5. Including sheared flow improves particle and energy confinement.

Results presented here show a fast-growing instability at smalla/r_(Li). The instability is identified as an electron drift mode asalso seen in earlier 5M2F modeling. Two important electron driftinstabilities (sometimes called microinstabilities) exist where theelectron cross-field drift speed (v_(dr)=|v_(e)−v_(i)|) approaches orexceeds v_(Ti): the lower-hybrid drift instability (LHDI) and the ionacoustic instability. In the 5M2F equilibria used here, the radiallyuniform electron drift speed is v_(dr)=j_(z)/(en). Using the equilibriumexpressions for j_(z) and n gives

$\begin{matrix}{{v_{dr} = {2{v_{Ti}\left( \frac{a}{r_{Li}} \right)}^{- 1}}},} & (38)\end{matrix}$where Eq. (26), evaluated at r=a, is used to replace I_(p) with peakB_(θ), and the definition of r_(Li) is applied. Thus, instability isexpected in the region with a/r_(Li)<2 such that v_(dr)>v_(Ti). Thisexpectation is consistent with the presence of the fast-growing modeseen in FIGS. 13A-14 at a/r_(Li)≤1.8. Most likely, the observedinstability is not LHDI, since LHDI occurs at wavelengths smaller thanr_(Le). At a/r_(Li)=1.8, r_(Le)/a≈0.013 (with m_(i)/m_(e)=1836). Theshortest perturbed wavelength considered has ka=80/3 and an associatedwavelength (λ/a=2π/ka≈0.24) that is ≈18 times larger than r_(Le).Instead of LHDI, the instability could be the ion acoustic instability.As discussed in, this instability is expected in a singly-ionized plasmaonly if T_(e)»T_(i). With T_(e)=T_(i) as in the plasmas considered here,the acoustic speed is comparable to the ion thermal velocity, andacoustic waves are Laundau damped. Because the 5M2F model does notcapture the Landau damping, it might admit an ion acoustic instabilityin regimes with T_(i)≈T_(e). In future work, behavior of the ionacoustic mode and LHDI in the 5M2F model may be considered in greaterdetail.

Modeling linear growth in the presence of positive sheared flow with the5M2F model at a/r_(Li)=2.357 gives results nearly identical to Hall-MHDmodeling. With either linear or parabolic sheared flow profiles,complete stabilization is projected to occur for v_(sf) ^(a)/v_(A)≈0.8.This result is consistent with PIC results for sheared flowstabilization, which indicated stabilization of the FuZE-like plasma atv_(sf) ^(a)/v_(A)=0.75. With negative sheared flow (i.e., ion flow zeroat r=0 and opposite the direction of current for r>0), the stabilizingeffect is weaker, and stabilization is projected to occur for v_(sf)^(a)/v_(A)>1. In SFS Z-pinch experiments, observed sheared flow speedsat the pinch edge during the stable plasma operation are typicallyv_(sf) ^(a)/v_(A)≈0.5. The gradient in velocity near the pinch edge issometimes observed to be steeper than even a parabolic profile, so it ispossible that such concentrated gradients could play a role in observedstability. But another possibility is that the experimental sheared flowwould not stabilize a Bennett profile; instead, observed m=0 stabilitymay occur because the profiles are relaxed such that they are(m=0)-stable with lower levels of shear.

The 5M2F model with real ion-electron mass ratio, with or withoutcorrected Braginskii transport, gives results that differ from PICresults, as shown in FIG. 22 . In particular, the 5M2F growth rate peakis higher by a factor of two, and the peak occurs at ka≈10 vs. ka≈5 inthe PIC results. The 5M2F results do reproduce growth rate rollover athigh k, and have qualitative similarity to the PIC results.Understanding the specific causes of the higher and shifted peak in 5M2Fmodeling would require further investigation, but kinetic effects seemto be at play. Gyrokinetic modeling shows close agreement with PIC, evenwith an electrostatic model and simplified gyrokinetic Poisson equationthat captures some but not all large-r_(Li) effects (e.g., gyro-orbitaveraging is not included). It is interesting to compare these resultswith analysis of the Kelvin-Helmholtz instability (KHI) with Hall MHD,ideal 5M2F, and continuum kinetic modeling. In that work, KHI growthrates with Hall MHD are higher than the kinetic results. The 5M2F growthrates are higher than kinetic, but much closer than the Hall-MHDresults.

Nonlinear modeling results shown in FIG. 3 indicate that with moderatesheared flow (parabolic with v_(sf) ^(a)/v_(A)=0.5), following initialinstability, the plasma establishes a quasi-stable configuration withmore than half of the original pinch mass and over 80% of the initialpinch energy. Further research is warranted to understand whether thisself-organization behavior might resemble actual experimental behavior.

Prior MHD modeling has indicated that sheared flow more effectivelystabilizes short-wavelength m=0 modes than long ones. Using the samesettings as for the ka=5 nonlinear simulations above, a simulation withka=5/3 and moderate parabolic sheared flow (v_(sf) ^(a)/v_(A)=0.5) isrun to explore the nonlinear confinement behavior of a longer-wavelengthinstability. The instability grows more quickly and mass and energy arelost earlier, but the total losses are comparable. In addition, torepresent a more realistic noisy Z-pinch plasma, a “multi-mode” case isrun with ka≤25. That is, the domain length is such that L_(z)/a=6π/5(capturing one wavelength of ka=5/3); each available mode up to ka=25(for which 15 wavelengths are captured) is perturbed. For each mode, thephase in Eqs. (31) and (32) is set with ϕ₀ chosen randomly, and with noradial dependence (ζ=0). The result is that mass and energy loss areincreased, perhaps due to mode coupling. In all of the cases, however,mass loss is <50% and energy loss <20%. The general picture remains thesame as for the ka=5 case: moderate sheared flow seems to facilitatedevelopment of a quasi-stable plasma with limited losses.

The modeling presented here has focused on Z-pinch equilibria usingBennett profiles with uniform initial temperature. Research thatconsiders other profiles, including profiles that match the bestavailable experimental data, can give different m=0 stability behavior,including a stabilizing effect due to guiding center drifts, even withzero bulk flow. It is also well known that profiles satisfying theKadomtsev criterion are MHD stable. The stability of MHD-stable profilesin non-MHD models, e.g., the 5M2F model, could be considered in futurework. Another consideration is that even Kadomtsev-stable profiles canbe unstable to so-called entropy modes, but for gases with realisticadiabatic coefficient (Γ≤5/3 for real gases with three or more degreesof freedom), entropy mode instability occurs only if temperatureprofiles are non-uniform. In recent modeling, entropy modes areconsidered in the special case of Γ>2, with uniform temperature inBennett profiles. 5M2F modeling of entropy mode behavior in MHD-stableprofiles with non-uniform temperature and Γ≤5/3 would be of interest.

The 5M2F model is presented and the ideal model is extended to include aBraginskii-based closure. The model is applied to study the m=0 Z-pinchinstability, focusing on initial conditions based on Bennett equilibria.

Ideal 5M2F modeling results are benchmarked against prior MHD andHall-MHD results. Growth rates closely match prior results with andwithout sheared flow. Complete stabilization occurs at edge flow speedsof ≤0.8v_(A). Going beyond the prior work, reversed linear sheared flowis studied, revealing a relatively slow decrease in growth rate withincreasing flow speed, and projected full stabilization at edge flowspeed of ≈1.5v_(A). Another intriguing feature of the 5M2F results isthat at a/r_(Li)≤1.8, an electron drift instability is seen andtentatively identified as an ion acoustic mode.

The 5M2F model is also benchmarked against recent PIC modelinginitialized with a FuZE-like Bennett equilibrium. Peak growth in the5M2F results is approximately twice as fast as in PIC (γτ_(A)≈1.5 vs.0.77), and peaks at k values around twice as large (ka≈10 vs. 5). The5M2F results resemble the PIC results in the sense that a peak in growthrate occurs at moderate k, and then falls at higher k. Applying theBraginskii-based transport model does not change this behavior. Moregenerally, exploration of Braginskii-based transport in the 5M2Fframework has provided insight into the related physics, includinggyroviscosity, diamagnetic heat flux, and ion gyrorelaxational effects.Electron gyrorelaxation is flagged as a potentially important effect forfuture consideration. Kinetic physics, beyond the 5M2F model even withBraginskii transport, seems responsible for the observed differences inm=0 stability seen in PIC modeling.

Nonlinear modeling of the m=0 instability in the FuZE-like plasma isdone using the Braginskii-based 5M2F model. A simulation with edgesheared flow speed of 0.5v_(A) shows instability, followed by nonlinearmixing due to the shear, and relaxation to a quasi-steady state. Lossesof pinch ion inventory and pinch thermal energy are limited to ≈30% and≈10%, respectively.

The 5M2F model provides a compelling platform for high-fidelitycomputational Z-pinch research. In future work, various equilibriumprofiles, besides the Bennett profile, may be considered; a profile ofspecial interest is the MHD-stable Kadomtsev profile with temperaturegradients that may activate entropy modes. Simulations of linear andnonlinear 3D Z-pinch dynamics including the m=1 instability are alsowithin reach. Finally, the ability of 5M2F modeling to capture electrondrift instabilities should be studied; accurate and efficient modelingof the associated microturbulence may yield critical insights related tocurrent profiles and axial thermal transport in reactor-grade Z-pinchplasmas.

For equilibria with sheared flow, true two-fluid equilibria are derived.The total ion axial velocity is determined as v_(iz)=v_(iz0)+v_(sf),where v_(sf)(r)=v_(sf) ^(a)r/a (linear) or v_(sf)(r)v_(sf) ^(a)r²/a²(parabolic). Here, v_(iz0)=j_(z)/(2en_(i)) is the shear-free ionvelocity, which provides half of the equilibrium current. Magnetic fieldand current profiles are unchanged from the shear-free equilibriumprofiles of Eqs. (26) and (27). Ion pressure and density are alsounchanged. Specifically, ion pressure is half of the total pressuregiven in Eq. (28). Ion temperature is uniform, and is a free parameter;herein, T_(i) is set in terms of a/r_(Li) as shown in Eq. (33). Thenn_(i)=p_(i)/(k_(B)T_(i)). Radial electric field is determined from Eq.(29) for ion momentum balance. The results for linear and parabolicsheared flow are

$\begin{matrix}{{E_{r}^{linear} = {\frac{\mu_{0}I_{p}}{2\pi}\frac{v_{sf}^{a}r^{2}}{a^{3} + {ar}^{2}}}},} & (39)\end{matrix}$ $\begin{matrix}{E_{r}^{parabolic} = {\frac{\mu_{0}I_{p}}{2\pi}{\frac{v_{sf}^{a}r^{3}}{a^{4} + {a^{2}r^{2}}}.}}} & (40)\end{matrix}$Gauss's Law, Eq. (7), is used to determine n_(e), giving

$\begin{matrix}{{n_{e}^{linear} = {n_{i} - {\frac{\mu_{0} \in_{0}I_{p}}{\pi q_{i}}\frac{v_{sf}^{a}\left( {{3a^{2}r^{2}} + r^{4}} \right)}{2{{ar}\left( {a^{2} + r^{2}} \right)}^{2}}}}},} & (41)\end{matrix}$ $\begin{matrix}{n_{e}^{parabolic} = {n_{i} - {\frac{\mu_{0} \in_{0}I_{p}}{\pi q_{i}}{\frac{v_{sf}^{a}\left( {{2a^{2}r^{3}} + r^{5}} \right)}{a^{2}{r\left( {a^{2} + r^{2}} \right)}^{2}}.}}}} & (42)\end{matrix}$Electron pressure is then determined from Eq. (30). Results are

$\begin{matrix}{{p_{e}^{linear} = {{p_{i} - {\frac{\mu_{0}I_{p}^{2}}{8\pi^{2}}\left( v_{sf}^{a} \right)^{2}}} \in_{0}{\mu_{0} \times \frac{{a^{2}r^{2}} + {\left( {a^{2} + r^{2}} \right)^{2}{\ln\left( \frac{a^{2}}{a^{2} + r^{2}} \right)}}}{{a^{2}\left( {a^{2} + r^{2}} \right)}^{2}}}}},} & (43)\end{matrix}$ $\begin{matrix}{p_{e}^{parabolic} = {{p_{i} + {\frac{\mu_{0}I_{p}^{2}}{4\pi^{2}}\left( v_{sf}^{a} \right)^{2}}} \in_{0}{\mu_{0} \times \text{ }{\frac{{r^{2}\left( {a^{4} + {\frac{3}{2}a^{2}r^{2}} + r^{4}} \right)} + {{a^{2}\left( {a^{2} + r^{2}} \right)}^{2}{\ln\left( \frac{a^{2}}{a^{2} + r^{2}} \right)}}}{{a^{4}\left( {a^{2} + r^{2}} \right)}^{2}}.}}}} & (44)\end{matrix}$

These pressures match the shear-free P_(e) at r=0, and deviate slightlyelsewhere. Electron temperature is determined as p_(e)/(n_(e)k_(B)).Electron axial velocity is determined by requiring thatj_(z)=v_(ez)n_(e)q_(e)+v_(iz)n_(i)q_(i) is satisfied.

-   -   Embodiments of the present disclosure can be described in view        of the following clauses:

1. A device, comprising:

-   -   a first electrode positioned to define an outer boundary of an        acceleration volume;    -   a second electrode positioned to define an inner boundary of the        acceleration volume;    -   at least one power supply to drive an electric current along a        Z-pinch plasma column between the first and second electrodes;    -   a set of valves comprising at least one gas-puff valve to        provide a neutral gas to the acceleration volume to fuel the        Z-pinch plasma column; and    -   a shaping part conductively connected to the second electrode        to, in a presence of the neutral gas provided by the at least        one gas-puff valve, cause a gas breakdown of the neutral gas to        generate a sheared flow velocity profile.

2. The device of clause 1, wherein an electron flow of the electriccurrent is from the second electrode to the first electrode.

3. The device of any one of clauses 1 and 2, wherein the shaping partincorporates at least one conductive ring comprising at least onecontact surface that is electrically connected to an outer surface ofthe second electrode.

4. The device of clause 3, wherein the at least one conductive ringincorporates conductive materials chemically and thermo-mechanicallycompatible with conductors of the second electrode, and a plasma-facingportion of the at least one shaping part incorporates at least onerefractory metal.

5. The device of clause 4, wherein the at least one refractory metalcomprises one or more of W, Ta, Nb, Mo, Re, Ti, V, Cr, Mn, Zr, Tc, Ru,Rh, Hf, Os, Ir, or an alloy of any one or more preceding metals.

6. The device of any one of clauses 4 and 5, wherein the plasma-facingportion incorporates at least one conductive form of carbon comprisingone or more of graphite, sintered carbon powders, pressed carbonpowders, carbon fiber, or carbon nanotube incorporating structures.

7. The device of any one of clauses 4 to 6, wherein the plasma-facingportion contains at least one textured surface formed to incorporate aplurality of localized concave elements forming structured arrays toenhance local electric fields and to facilitate electron field emission.

8. The device of clause 7, wherein the at least one textured surface hasbeen formed by a mechanical treatment comprising one or more of cutting,scratching, sanding, sandblasting, grooving, checkering, stumping,embossing, or knurling.

9. The device of any one of clauses 7 and 8, wherein the at least onetextured surface has been formed by a chemical treatment comprising oneor more of etching, chemical deposition, spraying, sputtering, ion andneutral implantation, or epitaxial growth.

10. The device of any one of clauses 1-9, wherein the set of valvesfurther comprises at least one plasma injector to provide an ionized gasto the acceleration volume to further fuel the Z-pinch plasma column.

11. The device of any one of clauses 1-10, wherein the second electrodeincorporates a conical electrode surface arranged to enhance momentumtransfer to ions and neutral particles in an axial direction of thefirst and second electrodes.

12. The device of any one of clauses 1-11, further comprising a thirdelectrode arranged between, and coaxially with respect to, the first andsecond electrodes, wherein the third electrode exhibits a taperedelectrode configuration and incorporates a conical electrode surfacearranged to enhance momentum transfer to ions and neutral particles inan axial direction of the first, second, and third electrodes.

13. A method, comprising:

-   -   activating one or more gas-puff valves to introduce an        axisymmetric volume of a neutral gas into an acceleration        volume;    -   generating a radial electric field to support a first current by        facilitating breakdown of the neutral gas, the first current        flowing, via the introduced neutral gas, between an inner        electrode and an outer electrode; and    -   forming a Z-pinch plasma column from the introduced neutral gas        to support a second current flowing between the inner electrode        and the outer electrode,    -   wherein the Z-pinch plasma column is surrounded and stabilized        by a sheared velocity plasma flow formed, at least in part, from        the neutral gas.

14. The method of clause 13, activating one or more plasma injectors tointroduce an axisymmetric volume of an ionized gas into the accelerationvolume.

15. The method of clause 14, wherein the axisymmetric volume of theionized gas is introduced to replenish the axisymmetric volume of theneutral gas following formation of the Z-pinch plasma column.

16. The method of any one of clauses 13-15, wherein the inner electrodeis an anode and the outer electrode is a cathode.

17. A plasma confinement system, comprising:

-   -   an outer electrode;    -   an inner electrode;    -   at least one power supply conductively coupled to each of the        inner and outer electrodes, terminals of the at least one power        supply configured to generate a potential difference between the        inner and outer electrodes; and    -   one or more first valves fluidly coupled to a fuel gas supply        and configured to direct sufficient neutral gas sourced from the        fuel gas supply to support a localized breakdown path between        the inner and outer electrodes and to establish a sheared        velocity plasma flow for a duration of a Z-pinch discharge        between the inner and outer electrodes.

18. The plasma confinement system of clause 17, wherein the inner andouter electrodes delimit an acceleration volume into which the neutralgas is directed by the one or more first valves.

19. The plasma confinement system of clause 17, further comprising anintermediate electrode,

-   -   wherein the inner and intermediate electrodes delimit an        acceleration volume into which the neutral gas is directed by        the one or more first valves.

20. The plasma confinement system of any one of clauses 17-19, furthercomprising one or more second valves fluidly coupled to the fuel gassupply and configured to direct sufficient ionized gas sourced from thefuel gas supply to maintain the sheared velocity plasma flow during theduration of the Z-pinch discharge.

21. A device, comprising:

-   -   a first electrode positioned to define an outer boundary of an        acceleration volume;    -   a second electrode arranged coaxially with respect to the first        electrode and positioned to define an inner boundary of the        acceleration volume;    -   at least one power supply to drive an electric current along a        Z-pinch plasma column between the first and second electrodes;        and    -   a set of valves to provide gas to the acceleration volume to        fuel the Z-pinch plasma column,    -   wherein an electron flow of the electric current is in a first        direction from the second electrode to the first electrode.

22. The device of clause 21, wherein the gas comprises a neutral gas,and

-   -   wherein the device further comprises a shaping part conductively        connected to the second electrode to, in a presence of the        neutral gas provided by the set of valves, cause a gas breakdown        of the neutral gas to generate a sheared flow velocity profile        in a second direction opposite to the first direction.

23. The device of clause 22, wherein the shaping part incorporates atleast one conductive ring comprising at least one contact surface thatis electrically connected to an outer surface of the second electrode.

24. The device of clause 23, wherein the at least one conductive ringincorporates conductive materials chemically and thermo-mechanicallycompatible with conductors of the second electrode, and a plasma-facingportion of the at least one shaping part incorporates at least onerefractory metal.

25. The device of clause 24, wherein the at least one refractory metalcomprises one or more of W, Ta, Nb, Mo, Re, Ti, V, Cr, Mn, Zr, Tc, Ru,Rh, Hf, Os, Ir, or an alloy of any one or more preceding metals.

26. The device of any one of clauses 24 and 25, wherein theplasma-facing portion incorporates at least one conductive form ofcarbon comprising one or more of graphite, sintered carbon powders,pressed carbon powders, carbon fiber, or carbon nanotube incorporatingstructures.

27. The device of any one of clauses 24-26, wherein the plasma-facingportion contains at least one textured surface formed to incorporate aplurality of localized concave elements forming structured arrays toenhance local electric fields and to facilitate electron field emission.

28. The device of clause 27, wherein the at least one textured surfacehas been formed by a mechanical treatment comprising one or more ofcutting, scratching, sanding, sandblasting, grooving, checkering,stumping, embossing, or knurling.

29. The device of any one of clauses 27 and 28, wherein the at least onetextured surface has been formed by a chemical treatment comprising oneor more of etching, chemical deposition, spraying, sputtering, ion andneutral implantation, or epitaxial growth.

30. The device of clause 21, wherein the gas is provided to theacceleration volume as an ionized gas.

31. The device of any one of clauses 21-30, wherein the second electrodeincorporates a conical electrode surface arranged to enhance momentumtransfer to ions and neutral particles in an axial direction of thefirst and second electrodes.

32. The device of any one of clauses 21-31, further comprising a thirdelectrode arranged between, and coaxially with respect to, the first andsecond electrodes, wherein the third electrode exhibits a taperedelectrode configuration and incorporates a conical electrode surfacearranged to enhance momentum transfer to ions and neutral particles inan axial direction of the first, second, and third electrodes.

33. A method, comprising:

-   -   activating one or more valves to introduce an axisymmetric        volume of a fuel gas into an acceleration volume; and    -   forming a Z-pinch plasma column from the introduced fuel gas to        support a Z-pinch current flowing between an inner anode and an        outer cathode surrounding an unsupported end of the inner anode,    -   wherein the Z-pinch plasma column is surrounded and stabilized        by a sheared velocity plasma flow formed from the fuel gas.

34. The method of clause 33, further comprising, prior to forming theZ-pinch plasma column, generating a radial electric field to support aninitial current flowing, via the introduced fuel gas, between the inneranode and the outer cathode.

35. The method of clause 34, wherein the fuel gas comprises a neutralgas, and

-   -   wherein the radial electric field supports the initial current        at least by facilitating breakdown of the neutral gas.

36. The method of any one of clauses 33-35, whereupon introduction ofthe fuel gas into the acceleration volume, the fuel gas comprises anionized gas.

37. A plasma confinement system, comprising:

-   -   an outer electrode;    -   an inner electrode concentrically positioned within the outer        electrode;    -   at least one power supply conductively coupled to each of the        inner and outer electrodes, terminals of the at least one power        supply oriented to generate a potential difference between the        inner and outer electrodes to drive electrons from the inner        electrode to the outer electrode; and    -   one or more valves fluidly coupled to a fuel gas supply and        configured to direct sufficient fuel gas sourced from the fuel        gas supply to drive a sheared velocity plasma flow for a        duration of a Z-pinch discharge between the inner and outer        electrodes.

38. The plasma confinement system of clause 37, wherein the inner andouter electrodes delimit an acceleration volume into which the fuel gasis directed by the one or more valves.

39. The plasma confinement system of clause 37, further comprising anintermediate electrode concentrically positioned between the inner andouter electrodes,

-   -   wherein the inner and intermediate electrodes delimit an        acceleration volume into which the fuel gas is directed by the        one or more valves.

40. The plasma confinement system of any one of clauses 37-39, whereinthe fuel gas comprises one or both of a neutral gas and an ionized gas.

While specific values, relationships, materials, and components havebeen set forth for purposes of describing concepts of the invention, itwill be appreciated by persons skilled in the art that numerousvariations and/or modifications may be made to the invention as shown inthe specific embodiments without departing from the spirit or scope ofthe basic concepts and operating principles of the invention as broadlydescribed. It should be recognized that, in the light of the aboveteachings, those skilled in the art can modify those specifics withoutdeparting from the invention taught herein. For example, numericalranges recited herein are exemplary and may be modified based on anoperating mode of a given plasma confinement system or based onmodifications to a size, function, configuration, etc. of the givenplasma confinement system. For instance, if the size of the given plasmaconfinement system increases, such ranges may scale proportionally(e.g., linearly, exponentially, etc.).

Having now fully set forth the embodiments and certain modifications ofthe concepts underlying the present invention, various other embodimentsas well as certain variations and modifications of the embodimentsherein shown and described may occur to those skilled in the art uponbecoming familiar with such underlying concepts. It is intended toinclude all such modifications, alternatives, and other embodimentsinsofar as they come within the scope of the appended claims orequivalents thereof. It should be understood, therefore, that theinvention may be practiced otherwise than as specifically set forthherein. Consequently, the present embodiments are to be considered inall respects as illustrative and not restrictive.

The specification and drawings are, accordingly, to be regarded in anillustrative rather than a restrictive sense. It will, however, beevident that various modifications and changes may be made thereuntowithout departing from the broader spirit and scope of the subjectmatter set forth in the claims.

Other variations are within the spirit of the present disclosure. Thus,while the disclosed techniques are susceptible to various modificationsand alternative constructions, certain illustrated embodiments thereofare shown in the drawings and have been described above in detail. Itshould be understood, however, that there is no intention to limit thesubject matter recited by the claims to the specific form or formsdisclosed but, on the contrary, the intention is to cover allmodifications, alternative constructions, and equivalents falling withinthe spirit and scope of this disclosure, as defined in the appendedclaims.

The use of the terms “a” and “an” and “the” and similar referents in thecontext of describing the disclosed embodiments (especially in thecontext of the following claims) are to be construed to cover both thesingular and the plural, unless otherwise indicated herein or clearlycontradicted by context. Similarly, use of the term “or” is to beconstrued to mean “and/or” unless contradicted explicitly or by context.The terms “comprising,” “having,” “including,” and “containing” are tobe construed as open-ended terms (i.e., meaning “including, but notlimited to,”) unless otherwise noted. The term “connected,” whenunmodified and referring to physical connections, is to be construed aspartly or wholly contained within, attached to, or joined together, evenif there is something intervening. Recitation of ranges of values hereinare merely intended to serve as a shorthand method of referringindividually to each separate value falling within the range, unlessotherwise indicated herein, and each separate value is incorporated intothe specification as if it were individually recited herein. The use ofthe term “set” (e.g., “a set of items”) or “subset” unless otherwisenoted or contradicted by context, is to be construed as a nonemptycollection comprising one or more members. Further, unless otherwisenoted or contradicted by context, the term “subset” of a correspondingset does not necessarily denote a proper subset of the correspondingset, but the subset and the corresponding set may be equal. The use ofthe phrase “based on,” unless otherwise explicitly stated or clear fromcontext, means “based at least in part on” and is not limited to “basedsolely on.”

Conjunctive language, such as phrases of the form “at least one of A, B,and C,” or “at least one of A, B and C,” (i.e., the same phrase with orwithout the Oxford comma) unless specifically stated otherwise orotherwise clearly contradicted by context, is otherwise understoodwithin the context as used in general to present that an item, term,etc., may be either A or B or C, any nonempty subset of the set of A andB and C, or any set not contradicted by context or otherwise excludedthat contains at least one A, at least one B, or at least one C. Forinstance, in the illustrative example of a set having three members, theconjunctive phrases “at least one of A, B, and C” and “at least one ofA, B and C” refer to any of the following sets: {A}, {B}, {C}, {A, B},{A, C}, {B, C}, {A, B, C}, and, if not contradicted explicitly or bycontext, any set having {A}, {B}, and/or {C} as a subset (e.g., setswith multiple “A”). Thus, such conjunctive language is not generallyintended to imply that certain embodiments require at least one of A, atleast one of B and at least one of C each to be present. Similarly,phrases such as “at least one of A, B, or C” and “at least one of A, Bor C” refer to the same as “at least one of A, B, and C” and “at leastone of A, B and C” refer to any of the following sets: {A}, {B}, {C},{A, B}, {A, C}, {B, C}, {A, B, C}, unless differing meaning isexplicitly stated or clear from context. In addition, unless otherwisenoted or contradicted by context, the term “plurality” indicates a stateof being plural (e.g., “a plurality of items” indicates multiple items).The number of items in a plurality is at least two but can be more whenso indicated either explicitly or by context.

Operations of processes described herein can be performed in anysuitable order unless otherwise indicated herein or otherwise clearlycontradicted by context. In an embodiment, a process such as thoseprocesses described herein (or variations and/or combinations thereof)is performed under the control of one or more computer systemsconfigured with executable instructions and is implemented as code(e.g., executable instructions, one or more computer programs or one ormore applications) executing collectively on one or more processors, byhardware or combinations thereof. In an embodiment, the code is storedon a computer-readable storage medium, for example, in the form of acomputer program comprising a plurality of instructions executable byone or more processors. In an embodiment, a computer-readable storagemedium is a non-transitory computer-readable storage medium thatexcludes transitory signals (e.g., a propagating transient electric orelectromagnetic transmission) but includes non-transitory data storagecircuitry (e.g., buffers, cache, and queues) within transceivers oftransitory signals. In an embodiment, code (e.g., executable code orsource code) is stored on a set of one or more non-transitorycomputer-readable storage media having stored thereon executableinstructions that, when executed (i.e., as a result of being executed)by one or more processors of a computer system, cause the computersystem to perform operations described herein. The set of non-transitorycomputer-readable storage media, in an embodiment, comprises multiplenon-transitory computer-readable storage media, and one or more ofindividual non-transitory storage media of the multiple non-transitorycomputer-readable storage media lack all of the code while the multiplenon-transitory computer-readable storage media collectively store all ofthe code. In an embodiment, the executable instructions are executedsuch that different instructions are executed by differentprocessors—for example, in an embodiment, a non-transitorycomputer-readable storage medium stores instructions and a main CPUexecutes some of the instructions while a graphics processor unitexecutes other instructions. In another embodiment, different componentsof a computer system have separate processors and different processorsexecute different subsets of the instructions.

Accordingly, in an embodiment, computer systems are configured toimplement one or more services that singly or collectively performoperations of processes described herein, and such computer systems areconfigured with applicable hardware and/or software that enable theperformance of the operations. Further, a computer system, in anembodiment of the present disclosure, is a single device and, in anotherembodiment, is a distributed computer system comprising multiple devicesthat operate differently such that the distributed computer systemperforms the operations described herein and such that a single devicedoes not perform all operations.

The use of any and all examples or exemplary language (e.g., “such as”)provided herein is intended merely to better illuminate variousembodiments and does not pose a limitation on the scope of the claimsunless otherwise claimed. No language in the specification should beconstrued as indicating any non-claimed element as essential to thepractice of inventive subject material disclosed herein.

Embodiments of this disclosure are described herein, including the bestmode known to the inventors for carrying out inventive conceptsdescribed herein. Variations of those embodiments may become apparent tothose of ordinary skill in the art upon reading the foregoingdescription. The inventors expect skilled artisans to employ suchvariations as appropriate, and the inventors intend for embodiments ofthe present disclosure to be practiced otherwise than as specificallydescribed herein. Accordingly, the scope of the present disclosureincludes all modifications and equivalents of the subject matter recitedin the claims appended hereto as permitted by applicable law. Moreover,any combination of the above-described elements in all possiblevariations thereof is encompassed by the scope of the present disclosureunless otherwise indicated herein or otherwise clearly contradicted bycontext.

All references including publications, patent applications, and patentscited herein are hereby incorporated by reference to the same extent asif each reference were individually and specifically indicated to beincorporated by reference and were set forth in its entirety herein.

The invention claimed is:
 1. A device, comprising: a cathode positionedto define an outer boundary of an acceleration volume; an anode arrangedcoaxially with respect to the cathode and positioned to define an innerboundary of the acceleration volume, the anode including one end atleast partially surrounded by the cathode; at least one power supply todrive an electric current along a Z-pinch plasma column and between thecathode and the anode; and a set of valves to provide gas to theacceleration volume to fuel the Z-pinch plasma column, wherein thecathode incorporates a tapered portion protruding towards the end of theanode at least partially surrounded by the cathode.
 2. The device ofclaim 1, wherein the gas comprises a neutral gas, and wherein the devicefurther comprises a shaping part conductively connected to the anode to,in a presence of the neutral gas provided by the set of valves, cause agas breakdown of the neutral gas to generate a sheared flow velocityprofile in a second direction opposite to the first direction.
 3. Thedevice of claim 2, wherein the shaping part incorporates at least oneconductive ring comprising at least one contact surface that iselectrically connected to an outer surface of the anode.
 4. The deviceof claim 3, wherein the at least one conductive ring incorporatesconductive materials chemically and thermo-mechanically compatible withconductors of the anode, and a plasma-facing portion of the at least oneshaping part incorporates at least one refractory metal.
 5. The deviceof claim 4, wherein the at least one refractory metal comprises one ormore of W, Ta, Nb, Mo, Re, Ti, V, Cr, Mn, Zr, Tc, Ru, Rh, Hf, Os, Ir, oran alloy of any one or more preceding metals.
 6. The device of claim 4,wherein the plasma-facing portion incorporates at least one conductiveform of carbon comprising one or more of graphite, sintered carbonpowders, pressed carbon powders, carbon fiber, or carbon nanotubeincorporating structures.
 7. The device of claim 4, wherein theplasma-facing portion contains at least one textured surface formed toincorporate a plurality of localized concave elements forming structuredarrays to enhance local electric fields and to facilitate electron fieldemission.
 8. The device of claim 7, wherein the at least one texturedsurface has been formed by a mechanical treatment comprising one or moreof cutting, scratching, sanding, sandblasting, grooving, checkering,stumping, embossing, or knurling.
 9. The device of claim 7, wherein theat least one textured surface has been formed by a chemical treatmentcomprising one or more of etching, chemical deposition, spraying,sputtering, ion and neutral implantation, or epitaxial growth.
 10. Thedevice of claim 1, wherein the gas is provided to the accelerationvolume as an ionized gas.
 11. The device of claim 1, wherein the anodeincorporates a conical electrode surface arranged to enhance momentumtransfer to ions and neutral particles in an axial direction of thecathode and the anode.
 12. The device of claim 1, further comprising anintermediate electrode arranged between, and coaxially with respect to,the cathode and the anode, wherein the intermediate electrode exhibits atapered electrode configuration and incorporates a conical electrodesurface arranged to enhance momentum transfer to ions and neutralparticles in an axial direction of the cathode, the anode, and theintermediate electrode.
 13. A method, comprising: activating one or morevalves to introduce an axisymmetric volume of a fuel gas into anacceleration volume; and forming a Z-pinch plasma column from theintroduced fuel gas to support a Z-pinch current flowing between aninner anode and an outer cathode, the outer cathode arranged coaxiallywith respect to the inner anode so as to surround an unsupported end ofthe inner anode and such that a tapered end of the outer cathodeprotrudes towards the unsupported end of the inner anode, wherein theZ-pinch plasma column is surrounded and stabilized by a sheared velocityplasma flow formed from the fuel gas.
 14. The method of claim 13,further comprising, prior to forming the Z-pinch plasma column,generating a radial electric field to support an initial currentflowing, via the introduced fuel gas, between the inner anode and theouter cathode.
 15. The method of claim 14, wherein the fuel gascomprises a neutral gas, and wherein the radial electric field supportsthe initial current at least by facilitating breakdown of the neutralgas.
 16. The method of claim 13, whereupon introduction of the fuel gasinto the acceleration volume, the fuel gas comprises an ionized gas. 17.A plasma confinement system, comprising: an outer cathode; an inneranode concentrically positioned within the outer cathode and arrangedcoaxially with respect to the outer cathode; at least one power supplyconductively coupled to each of the inner anode and the outer cathode,terminals of the at least one power supply oriented to generate apotential difference between the inner anode and the outer cathode; andone or more valves fluidly coupled to a fuel gas supply and configuredto direct sufficient fuel gas sourced from the fuel gas supply to drivea sheared velocity plasma flow for a duration of a Z-pinch dischargeflowing between the inner anode and the outer cathode, wherein the outercathode comprises a tapered electrode surface arranged coaxially withrespect to the inner anode so as to reduce a distance of the Z-pinchdischarge flowing between the inner anode and the outer cathode.
 18. Theplasma confinement system of claim 17, wherein the inner anode and theouter cathode delimit an acceleration volume into which the fuel gas isdirected by the one or more valves, and wherein the outer cathodecomprises a plurality of portions electrically connected to one another.19. The plasma confinement system of claim 17, further comprising anintermediate electrode concentrically positioned between the inner anodeand the outer cathode, wherein the inner anode and the intermediateelectrode delimit an acceleration volume into which the fuel gas isdirected by the one or more valves.
 20. The plasma confinement system ofclaim 17, wherein the fuel gas comprises one or both of a neutral gasand an ionized gas, and wherein the inner anode comprises a non-concaveelectrode surface facing the tapered electrode surface of the outercathode.